# Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

Authors: Siegfried Carl; Dumitru Motreanu
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

The study of multiple solutions for quasilinear
elliptic problems under Dirichlet or nonlinear Neumann type
boundary conditions has received much attention over the last
decades. The main goal of this paper is to present multiple
solutions results for elliptic inclusions of Clarke's gradient
type under Dirichlet boundary condition involving thep-Laplacian which, in general, depend on two parameters.
Assuming different structure and smoothness assumptions on the
nonlinearities generating the multivalued term, we prove the
existence of multiple constant-sign and sign-changing (nodal)
solutions for parameters specified in terms of the Fučik
spectrum of the p-Laplacian. Our approach will be based on
truncation techniques and comparison principles (sub-supersolution
method) for elliptic inclusions combined with variational and
topological arguments for, in general, nonsmooth functionals, such
as, critical point theory, Mountain Pass Theorem, Second
Deformation Lemma, and the variational characterization of the
“beginning”of the Fu\v cik spectrum of the p-Laplacian. In particular, the existence of extremal constant-sign solutions and
their variational characterization as global (resp., local) minima
of the associated energy functional will play a key-role in the
proof of sign-changing solutions.


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## Body

## 1. Introduction

LetΩ⊂ℝN be a bounded domain with a C2-boundary ∂Ω, and let V=W1,p(Ω) and V0=W01,p(Ω), 1<p<+∞, denote the usual Sobolev spaces with their dual spaces V* and V0*, respectively. We consider the following nonlinear multi-valued elliptic boundary value problem under Dirichlet boundary condition: find u∈V0∖{0} and parameters a∈ℝ, b∈ℝ such that(1.1)-Δpu∈∂j(·,u,a,b)inV0*,
where Δpu=div(|∇u|p-2∇u) is the p-Laplacian, and s↦∂j(x,s,a,b) denotes Clarke's generalized gradient of some locally Lipschitz function s↦j(x,s,a,b) which depends on x∈Ω and the parameters a,b. For a=b=:λ problem (1.1) reduces to(1.2)-Δpu∈∂j(·,u,λ)inV0*,
which may be considered as a nonlinear and nonsmooth eigenvalue problem. We are going to study the existence of multiple solutions of (1.1) for two different classes of j which are in some sense complementary. Our presentation is based on and extends the authors' recent results obtained in [1–3]. For the first class of j we let a=b=λ and assume the following structure of j:(1.3)j(x,s,λ)=∫0sf(x,t,λ)dt,
where f:Ω×ℝ×(0,λ¯)→ℝ is such that f(·,·,λ):Ω×ℝ→ℝ is a Carathéodory function. Problem (1.1) reduces then to the following nonlinear eigenvalue problem:(1.4)u∈V0∖{0}:-Δpu=f(·,u,λ)inV0*,
which will be considered in Section 2 when the parameter λ is small enough.The second class ofj has the following structure:(1.5)j(x,s,a,b)=ap(s+)p+bp(s-)p+G(x,s),
where s+=max{s,0} and s-=max{-s,0} is the positive and negative part of s, respectively, and G:Ω×ℝ→ℝ is assumed to be the primitive of a measurable function g:Ω×ℝ→ℝ that is merely bounded on bounded sets; that is, g∈Lloc∞(Ω×ℝ) and G is given by (1.6)G(x,s):=∫0sg(x,t)dt.
Problem (1.1) reduces then to the following parameter-dependent multi-valued elliptic problem:(1.7)u∈V0∖{0}:-Δpu∈a(u+)p-1-b(u-)p-1-∂G(x,u)inV0*,
which will be studied in Section 3 for parameters a and b large enough. Note that s↦∂G(x,s) stands for the generalized Clarke's gradient of the locally Lipschitz function s↦G(x,s). Obviously, if g:Ω×ℝ→ℝ is a Carathéodory function, that is, x↦g(x,s) is measurable in Ω for all s∈ℝ and s↦g(x,s) is continuous in ℝ for a.a. x∈Ω, then ∂G(x,s)={g(x,s)} is single-valued, and thus problem (1.7) reduces to the following nonlinear elliptic problem depending on parameters a and b: find u∈V0∖{0} and constants a∈ℝ, b∈ℝ such that(1.8)-Δpu=a(u+)p-1-b(u-)p-1-g(x,u)inV0*.
Multiple solution results for (1.8) were obtained by the authors in [4]. Furthermore, note that (1.9)|u|p-2u=|u|p-2(u+-u-)=(u+)p-1-(u-)p-1.
Therefore, if one assumes, in addition, a=b=:λ, then (1.8) reduces to the nonlinear elliptic eigenvalue problem: find u∈V0∖{0} and a constant λ∈ℝ such that(1.10)-Δpu=λ|u|p-2u-g(x,u)inV0*.
In a recent paper (see [5]) the authors considered the eigenvalue problem (1.10) for a Carathéodory function g. Combining the method of sub-supersolution with variational techniques and assuming certain growth conditions of s↦g(x,s) at infinity and at zero the authors were able to prove the existence of at least three nontrivial solutions including one that changes sign. The results in [5] improve among others recent results obtained in [6]. For a=b=:λ, (1.7) reduces to the corresponding multivalued eigenvalue problem: find u∈V0∖{0} and a constant λ∈ℝ such that(1.11)-Δpu∈λ|u|p-2u-∂G(x,u)inV0*.
The existence of multiple solutions for (1.11) has been shown recently in [7] where techniques for single-valued problems developed in [5] and hemivariational methods applied in [8] have been used. Multiplicity results for (1.11) have been obtained also in [9].The existence of multiple solutions for semilinear and quasilinear elliptic problems has been studied by a number of authors, for example, [10–24]. All these papers deal with nonlinearities (x,s)↦g(x,s) that are sufficiently smooth.


## 2. Problem (1.4) for λ being Small

The aim of this section is to provide an existence result of multiple solutions for all values of the parameterλ in an interval (0,λ0), with λ0>0, guaranteeing that for any such λ there exist at least three nontrivial solutions of problem (1.4), two of them having opposite constant sign and the third one being sign-changing (or nodal). More precisely, we demonstrate that under suitable assumptions there exist a smallest positive solution, a greatest negative solution, and a sign-changing solution between them, whereas the notions smallest andgreatest refer to the underlying natural partial ordering of functions. This continues the works of Jin [25] (where p=2 and f(x,s,λ) is Hölder continuous with respect to (x,s)∈Ω¯×ℝ for every fixed λ) and of Motreanu-Motreanu-Papageogiou [26]. In these cited works one obtains three nontrivial solutions, two of which being of opposite constant sign, but without knowing that the third one changes sign. Here we derive the new information of having, in addition, a sign-changing solution by strengthening the unilateral condition for the right-hand side of the equation in (1.4) at zero. Furthermore, under additional hypotheses, we demonstrate that one can obtain two sign-changing solutions.


### 2.1. Hypotheses and Example

LetLq(Ω)+,1≤q≤+∞, denote the positive cone of Lq(Ω) given by (2.1)Lq(Ω)+={v∈Lq(Ω):v(x)≥0fora.a.x∈Ω}.
We impose the following hypotheses on the nonlinearity f(x,s,λ) in problem (1.4).H(f)
f:Ω×ℝ×(0,λ̅)→ℝ, with λ̅>0, is a function such that f(x,0,λ)=0 for a.a. x∈Ω, whenever λ∈(0,λ̅), and one has the following.(i)
For allλ∈(0,λ̅), f(·,·,λ) is Carathéodory (i.e., f(·,s,λ) is measurable for all s∈ℝ and f(x,·,λ) is continuous for almost all x∈Ω).(ii)
There are constantsc>0, r>p-1, and functions a(·,λ)∈L∞(Ω)+ (λ∈(0,λ̅)) with ∥a(·,λ)∥∞→0 as λ↓0 such that
(2.2)|f(x,s,λ)|≤a(x,λ)+c|s|rfora.a.x∈Ω∀(s,λ)∈ℝ×(0,λ̅).(iii)
For allλ∈(0,λ̅) there exist constants μ0=μ0(λ)>λ2, ν0=ν0(λ)>μ0 and a set Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.3)μ0<lim infs→0f(x,s,λ)|s|p-2s≤lim sups→0f(x,s,λ)|s|p-2s≤ν0
uniformly with respect to x∈Ωλ.InH(f)(iii), λ2 denotes the second eigenvalue of (-Δp,V0). As mentioned in the Introduction, the strengthening with respect to [26] (see also [25]) of the unilateral condition for the right-hand side f in (1.4), which enables us to obtain, in addition, sign-changing solutions, consists in adding the part involving the limit superior in H(f)(iii).Let us provide an example where all the assumptions formulated inH(f) are fulfilled.Example 2.1.
For the sake of simplicity we drop thex dependence for the function f in the right-hand side of (1.4). The function f:ℝ×(0,+∞)→ℝ given by
(2.4)f(s,λ)=λarctan(λ+λ2λ|s|p-2s)+c|s|r-1s∀(s,λ)∈ℝ×(0,+∞),
with c>0 and r>p-1, satisfies hypotheses H(f). Next we give an example of function f:ℝ×(0,+∞)→ℝ verifying assumptions H(f) which is generally not odd with respect to s:
(2.5)f(s,λ)={λarctan(a1λ+λ2λ|s|p-2s)+c1|s|r1-1sifs≤0,λarctan(a2λ+λ2λsp-1)+c2sr2ifs>0,
with λ>0, a1≥1, a2≥1, c1>0, c2>0, r1>p-1, r2>p-1.


### 2.2. Constant-Sign Solutions

The operator-Δp:V0→V0* is maximal monotone and coercive; therefore there exists a unique solution e∈V0 of the Dirichlet problem(2.6)e∈V0:-Δpe=1inV0*.
With s-=max{-s,0} for s∈ℝ, and using -e-∈V0 as a test function, we see that(2.7)∥∇e-∥pp=〈-Δpe,-e-〉=-∫Ωe-(x)dx≤0,
which implies that e≥0. From the nonlinear regularity theory (cf., e.g., [27, Theorem 1.5.6]) we have e∈C01(Ω¯). Then from the nonlinear strong maximum principle (see [28]) we infer that e∈int(C01(Ω¯)+). Here int(C01(Ω¯)+) denotes the interior of the positive cone C01(Ω¯)+={u∈C01(Ω¯):u(x)≥0,∀x∈Ω} in the Banach space C01(Ω¯)={u∈C1(Ω¯):u(x)=0,∀x∈∂Ω}, given by(2.8)int(C01(Ω¯)+)={u∈C01(Ω¯):u(x)>0,∀x∈Ω,and∂u∂n(x)<0,∀x∈∂Ω},
where n=n(x) is the outer unit normal at x∈∂Ω.Lemma 2.2.
Let the datac, r, and a(·,λ) be as in H(f)(ii). Then for every constant θ>0 there is λ0∈(0,λ̅) with the property that if λ∈(0,λ0), one can choose ξ0=ξ0(λ)∈(0,θ) such that
(2.9)c(ξ0∥e∥∞)r+∥a(·,λ)∥∞<ξ0p-1.Proof.
On the contrary there would exist a constantθ>0 and a sequence λn↓0 as n→∞ such that
(2.10)c(ξ∥e∥∞)r+∥a(·,λn)∥∞≥ξp-1∀n∈ℕ,ξ∈(0,θ).
Letting n→∞ we get c∥e∥∞rξr-p+1≥1 for all ξ∈(0,θ) because we have ∥a(·,λ)∥∞→0 as λ↓0. Since r>p-1, a contradiction is achieved as ξ↓0. Therefore (2.9) holds true.We denote byλ1 the first eigenvalue of (-Δp,V0) and by φ1 the eigenfunction of (-Δp,V0) corresponding to λ1 satisfying (2.11)φ1∈int(C01(Ω¯)+),∥φ1∥p=1.Lemma 2.3.
AssumeH(f)(i) and (ii) and the following weaker form of hypothesis H(f)(iii): for all λ∈(0,λ̅) there exist μ0=μ0(λ)>λ1 and Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.12)μ0<lim infs→0f(x,s,λ)|s|p-2s
uniformly with respect to x∈Ωλ.
Fix a constantθ>0 and consider the corresponding number λ0∈(0,λ̅) obtained in Lemma 2.2. Then for any λ∈(0,λ0) the function u¯=ξ0e∈int(C01(Ω¯)+), with ξ0∈(0,θ) given by Lemma 2.2, is a supersolution for problem (1.4), and the function u̲=εφ1∈int(C01(Ω¯)+) is a subsolution of problem (1.4) provided that the number ε>0 is sufficiently small.Proof.
For a fixedλ∈(0,λ0), from (2.9) and H(f)(ii) we derive
(2.13)-Δpu¯=ξ0p-1>∥a(·,λ)∥∞+c∥u¯∥∞r≥f(·,u¯(·),λ),
which says that u¯=ξ0e is a supersolution for problem (1.4).
On the other hand, by hypothesis we can findμ=μ(λ)>λ1 and δ=δ(λ)>0 such that(2.14)μ<f(x,s,λ)|s|p-2sfora.a.x∈Ω∀0<|s|≤δ.
Choose ε∈(0,δ/∥φ1∥∞). Then by (2.14) we have
(2.15)-Δp(εφ1)=λ1εp-1φ1p-1<μεp-1φ1p-1<f(x,εφ1(x),λ)fora.a.x∈Ω,
which ensures that u̲=εφ1 is a subsolution of problem (1.4).The following result which asserts the existence of two solutions of problem (1.4) having opposite constant sign and being extremal plays an important role in the proof of the existence of sign-changing solutions.Theorem 2.4.
AssumeH(f)(i) and (ii) and the following weaker form of H(f)(iii): for all λ∈(0,λ̅) there exist constants μ0=μ0(λ)>λ1, ν0=ν0(λ)>μ0 and a set Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.16)μ0<lim infs→0f(x,s,λ)|s|p-2s≤lim sups→0f(x,s,λ)|s|p-2s≤ν0
uniformly with respect to x∈Ωλ. Then for all b>0 there exists a number λ0∈(0,λ̅) with the property that if λ∈(0,λ0), then there is a constant ξ0=ξ0(λ)∈(0,b/∥e∥∞) such that problem (1.4) has a least positive solution u+=u+(λ)∈int(C01(Ω¯)+) in the order interval [0,ξ0e] and a greatest negative solution u-=u-(λ)∈-int(C01(Ω¯)+) in the order interval [-ξ0e,0].Proof.
Since the proof of the existence of the greatest negative solution follows the same lines, we only provide the arguments for the existence of the least positive solution.
Applying Lemma2.3 for θ=b/∥e∥∞we find λ0∈(0,λ̅) as therein. Fix λ∈(0,λ0). Lemma 2.3 ensures that u¯=ξ0e∈int(C01(Ω¯)+) is a supersolution for problem (1.4), with ξ0∈(0,b/∥e∥∞) given by Lemma 2.2, and u̲=εφ1∈int(C01(Ω¯)+) is a subsolution for problem (1.4) if ε>0 is small enough. Passing eventually to a smaller ε>0, we may assume that εφ1≤ξ0e. Then by the method of sub-supersolution we know that in the order interval [εφ1,ξ0e] there is a least (i.e., smallest) solution uε=uε(λ)∈int(C01(Ω¯)+) of problem (1.4) (see [29]).
We thus obtain that for every positive integern sufficiently large there is a least solution un∈int(C01(Ω¯)+) of problem (1.4) in the order interval [(1/n)φ1,ξ0e]. Clearly, we have(2.17)un↓u+pointwise,
with some function u+:Ω→ℝ satisfying 0≤u+≤ξ0e. First we claim that
(2.18)u+isasolutionofproblem(1.4).
Taking into account that un solves (1.4), and the fact that un belongs to the order interval [0,ξ0e], from H(f)(ii) we see that
(2.19)∥∇un∥pp=∫Ωf(x,un(x),λ)un(x)dx≤∫Ω(a(x,λ)+cξ0re(x)r)ξ0e(x)dx,
which implies the boundedness of the sequence (un) in V0. Then due to (2.17) we have that u+∈V0 as well as
(2.20)un⇀u+inV0,un→u+inLp(Ω)anda.e.inΩ.
Since un solves problem (1.4), one has
(2.21)〈-Δpun,φ〉=∫Ωf(x,un(x),λ)φ(x)dx,∀φ∈V0.
Setting φ=un-u+ in (2.21) gives
(2.22)〈-Δpun,un-u+〉=∫Ωf(x,un(x),λ)(un(x)-u+(x))dx.
As already noticed that the sequence (f(·,un(·),λ) is uniformly bounded on Ω, so (2.20) and (2.22) yield
(2.23)limn→∞〈-Δpun,un-u+〉=0.
The S+-property of -Δp on V0 implies
(2.24)un→u+inV0asn→∞.
The strong convergence in (2.24) and Lebesgue's dominated convergence theorem permit to pass to the limit in (2.21) that results in (2.18).
By (2.18) and the nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) it turns out u+∈C01(Ω¯). The choice of ξ0 guarantees that(2.25)0≤u+(x)≤ξ0e(x)≤bfora.e.x∈Ω.
Thus, from (2.18), assumptions H(f)(ii) and (iii), and the boundedness of u+, we get
(2.26)-Δpu+(x)=f(x,u+(x),λ)≥-ĉu+(x)p-1fora.a.x∈Ω,
with a constant ĉ>0. Applying the nonlinear strong maximum principle (cf. [28]) we conclude that either u+=0 or u+∈int(C01(Ω¯)+).
We claim that(2.27)u+∈int(C01(Ω¯)+).
Assume on the contrary that u+=0. Then (2.17) becomes
(2.28)un(x)↓0∀x∈Ω.
Since un≥(1/n)φ1, we may consider
(2.29)ũn=un∥∇un∥p∀n.
Along a relabelled subsequence we may suppose
(2.30)ũn⇀ũinV0,ũn→ũinLp(Ω)anda.e.inΩ
for some ũ∈V0. Moreover, one can find a function w∈Lp(Ω)+ such that |ũn(x)|≤w(x) for almost all x∈Ω. Relation (2.21) reads
(2.31)〈-Δpũn,φ〉=∫Ωf(x,un(x),λ)∥∇un∥pp-1φdx,∀φ∈V0.
Setting φ=ũn-ũ leads to
(2.32)〈-Δpũn,ũn-ũ〉=∫Ωf(x,un(x),λ)∥∇un∥pp-1(ũn-ũ)dx.
By H(f)(iii) we know that there exist constants c0=c0(λ)>λ1 and α=α(λ)>0 such that
(2.33)|f(x,s,λ)|≤c0|s|p-1fora.a.x∈Ω,∀|s|<α,
while H(f)(ii) entails
(2.34)|f(x,s,λ)|≤a(x,λ)+c|s|r≤(∥a(·,λ)∥∞αr+c)|s|r
for a.a. x∈Ω and for all |s|≥α. Combining the two estimates gives
(2.35)|f(x,s,λ)|≤c0|s|p-1+c1|s|rfora.a.x∈Ω,∀s∈ℝ
with a constant c1=c1(λ)>0. Since un∈[(1/n)φ1,ξ0e], r>p-1 and (2.35) holds, there exists a constant C>0 such that
(2.36)|f(x,un(x),λ)|un(x)p-1≤Cfora.a.x∈Ω,∀n.
We see from (2.36) that
(2.37)|f(x,un(x),λ)|∥∇un∥pp-1|ũn(x)-ũ(x)|≤Cw(x)p-1(w(x)+|ũ(x)|)fora.a.x∈Ω.
Then, because the right-hand side of the above inequality is in L1(Ω), by means of (2.30) and (2.36) we can apply Lebesgue's dominated convergence theorem to get
(2.38)limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1(ũn-ũ)dx=0.
Consequently, from (2.32) we obtain
(2.39)limn→∞〈-Δpũn,ũn-ũ〉=0.
The S+-property of -Δp on V0 implies
(2.40)ũn→ũinV0asn→∞.
On the basis of (2.31) and (2.40) it follows
(2.41)〈-Δpũ,φ〉=limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1φdx,∀φ∈V0.
Notice from (2.36) that
(2.42)|f(x,un(x),λ)|∥∇un∥pp-1|φ(x)|≤Cw(x)p-1|φ(x)|
for a.a. x∈Ω and for all φ∈V0. We are thus allowed to apply Fatou's lemma which in conjunction with (2.28), (2.30), and (2.16) ensures
(2.43)limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1φ(x)dx=limn→∞∫Ωf(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x)dx≥∫Ωlim infn→∞(f(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x))dx≥μ0∫Ωũ(x)p-1φ(x)dx
for all φ∈V0,+:=V0∩Lp(Ω)+. Thanks to (2.41) we obtain
(2.44)〈-Δpũ,φ〉≥μ0∫Ωũ(x)p-1φ(x)dx,∀φ∈V0,+.
Owing to (2.42) we may once again use Fatou's lemma; so according to (2.28), (2.30), and the last part of (2.16), we find
(2.45)limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1φ(x)dx=limn→∞∫Ωf(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x)dx≤∫Ωlim supn→∞(f(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x))dx≤ν0∫Ωũ(x)p-1φ(x)dx
for all φ∈V0,+. Then (2.41) ensures
(2.46)〈-Δpũ,φ〉≤ν0∫Ωũ(x)p-1φ(x)dx,∀φ∈V0,+.
Combining (2.44) and (2.46) results in
(2.47)μ0ũp-1≤-Δpũ≤ν0ũp-1a.e.inΩ,
which guarantees to have ũ∈L∞(Ω) (see [27, Theorem 1.5.5]). Since by (2.47) we know that Δpũ∈L∞(Ω), we are in a position to address Theorem 1.5.6 in [27], which provides ũ∈C1,β(Ω¯) with some β∈(0,1). This regularity up to the boundary and the fact that ũ≥0 a.e. in Ω and (2.47) enable us to refer to the strong maximum principle (see Theorem 5 of Vázquez [28]). Recalling that ũ does not vanish identically on Ω (because ∥∇ũ∥p=1) we deduce that ũ(x)>0 for all x∈Ω and (∂ũ/∂n)(x)<0 for all x∈∂Ω which amounts to saying ũ∈int(C01(Ω¯)+). Consequently, there exist constants k0>0 and k1>0 such that
(2.48)k0φ1≤ũ<k1φ1a.e.inΩ.
Following [30] let us denote
(2.49)I(u,v)=〈-Δpu,up-vpup-1〉-〈-Δpv,up-vpvp-1〉
whenever (u,v)∈DI, where
(2.50)DI={(w1,w2)∈(V0)2:wi≥0,wiwj∈L∞(Ω)fori,j∈{1,2}}.
Relation (2.48) justifies that (k1φ1,ũ)∈DI. Then Proposition 1 of Anane [30] implies I(k1φ1,ũ)≥0. On the other hand a direct computation based on (2.48) and (2.47) shows
(2.51)I(k1φ1,ũ)=〈-Δp(k1φ1),(k1φ1)p-ũp(k1φ1)p-1〉-〈-Δpũ,(k1φ1)p-ũpũp-1〉≤(λ1-μ0)∫Ω((k1φ1)p-ũp)dx<0.
This contradiction proves that the claim in (2.27) holds true.
In view of (2.18) it remains to establish that u+ is the smallest positive solution of problem (1.4) in the interval [0,u¯]. Let u∈V0 be a positive solution to (1.4) in [0,u¯]. Since u∈L∞(Ω), then (1.4) and H(f)(ii) allow to deduce that -Δpu∈L∞(Ω). Using Theorem 1.5.6 of [27] leads to u∈C01(Ω¯). Then, as u is a solution to (1.4) and u∈[0,u¯], with ∥u¯∥∞<b, by means of hypotheses H(f)(ii) and (iii), we are able to apply the strong maximum principle. So we get u∈int(C01(Ω¯)+), hence u∈[(1/n)φ1,u¯] for n sufficiently large. The fact that un is the least solution of (1.4) in [(1/n)φ1,u¯] ensures un≤u. Taking into account (2.17), we obtain u+≤u. This completes the proof.


### 2.3. Sign-Changing Solution

The main result of this section is as follows.Theorem 2.5.
Under hypothesesH(f), for all b>0, there exists a number λ0∈(0,λ̅) with the property that if λ∈(0,λ0), then problem (1.4) has a (positive) solution u+=u+(λ)∈int(C01(Ω¯)+), a (negative) solution u-=u-(λ)∈-int(C01(Ω¯)+), and a nontrivial sign-changing solution u0=u0(λ)∈C01(Ω¯) satisfying ∥u+∥∞<b, ∥u-∥∞<b, ∥u0∥∞<b.Proof.
Letb>0. Consider the positive number λ0 given by Theorem 2.4 and fix λ∈(0,λ0). Let u+∈int(C01(Ω¯)+) and u-∈-int(C01(Ω¯)+) be the two extremal solutions determined in Theorem 2.4. We introduce on Ω×ℝ the truncation functions
(2.52)τ+(x,s)={0ifs≤0,sif0<s<u+(x),u+(x)ifs≥u+(x),τ-(x,s)={u-(x)ifs≤u-(x),sifu-(x)<s<0,0ifs≥0,τ0(x,s)={u-(x)ifs≤u-(x),sifu-(x)<s<u+(x),u+(x)ifs≥u+(x)
and then define the following associated functionals:
(2.53)E+(u)=1p∥∇u∥pp-∫Ω∫0u(x)f(x,τ+(x,s),λ)dsdx,∀u∈V0,E-(u)=1p∥∇u∥pp-∫Ω∫0u(x)f(x,τ-(x,s),λ)dsdx,∀u∈V0,E0(u)=1p∥∇u∥pp-∫Ω∫0u(x)f(x,τ0(x,s),λ)dsdx,∀u∈V0.
It is clear that E+,E-,E0∈C1(V0).
We observe that ifv is a critical point of E+, then(2.54)〈-Δpv+Δpu+,(v-u+)+〉=∫Ω(f(x,τ+(x,v(x)),λ)-f(x,u+(x),λ))(v-u+)+dx=0
which implies v≤u+. Similarly, it follows that v≥0. This leads to
(2.55)visacriticalpointofE+⇒0≤v(x)≤u+(x)fora.a.x∈Ω.
Since the functionE+ is coercive and weakly lower semicontinuous, there exists a global minimizer z+∈V0 of it. Using (2.14), it is seen that(2.56)E+(z+)=infV0E+<0,
and so z+≠0. Relation (2.55) shows that z+ is a nontrivial solution of problem (1.4) belonging to the order interval [0,u+]. Via assumptions H(f)(ii) and (iii) and the boundedness of z+, we may apply the strong maximum principle which ensures z+>0 on Ω. In view of the minimality property of u+ as stated in Theorem 2.4, it follows that z+=u+. In fact, u+ is the unique global minimizer of E+.
Sinceu+∈int(C01(Ω¯)+), there exists a neighborhood 𝒰 of u+ in the space C01(Ω¯) such that 𝒰⊂C01(Ω¯)+. Therefore E0=E+ on 𝒰, which guarantees that u+ is a local minimizer of E0 on C01(Ω¯). It results that u+ is also a local minimizer of E0 on the space V0 (see [27], pages 655-656 ). Employing the functional E- and proceeding as in the case of u+, we establish that u- is a local minimizer of E0 on V0.
As in the case of (2.55), we verify that every critical point of E0 belongs to the set {u∈V0:u-(x)≤u(x)≤u+(x)a.e.x∈Ω}, which implies that every critical point of E0 is a solution to problem (1.4). The functional E0 is coercive, weakly lower semicontinuous, with infV0E0<0. Thus E0 has a global minimizer y0∈V0 with y0≠0. The above properties ensure that y0 is a nontrivial solution of problem (1.4) belonging to the order interval [u-,u+]. Assume y0≠u+ and y0≠u-. We claim that y0 changes sign. Indeed, if not, y0 would have constant sign, for instance y0≥0 a.e. on Ω. Using assumptions H(f)(ii) and (iii) and the boundedness of y0, we may apply the strong maximum principle which leads to y0>0 on Ω. This is impossible because it contradicts the minimality property of the solution u+ as given by Theorem 2.4. According to the claim, we obtain the conclusion of the theorem setting u0=y0.
Thus, the proof reduces to consider the casesy0=u+ or y0=u-. To make a choice, suppose y0=u+. We may also admit that u- is a strict local minimizer of E0. This is true since on the contrary we would find (infinitely many) critical points x0 of E0 belonging to the order interval [u-,u+] which are different from 0, u-, u+, and if x0 does not change sign, taking into account the strong maximum principle, the extremality properties of the solutions u-, u+ given in Theorem 2.4 will be contradicted. A straightforward argument allows then to find ρ∈(0,∥u+-u-∥) such that(2.57)E0(u+)≤E0(u-)<inf{E0(u):u∈∂Bρ(u-)},
where ∂Bρ(u-)={u∈V0:∥u-u-∥=ρ}. Relation (2.57) in conjunction with the Palais-Smale condition (which holds for E0 due to its coercivity) enables us to apply the mountain pass theorem to the functional E0 (see, e.g., [31]). In this way we get u0∈V0 satisfying E′0(u0)=0 and
(2.58)inf{E0(u):u∈∂Bρ(u-)}≤E0(u0)=infγ∈Γmaxt∈[-1,1]E0(γ(t)),
where
(2.59)Γ={γ∈C([-1,1],V0):γ(-1)=u-,γ(1)=u+}.
We infer from (2.57) and (2.58) that u0≠u-and u0≠u+.
The next step in the proof is to show that(2.60)E0(u0)<0.
By the equality in (2.58), it suffices to produce a path γ̂∈Γ such that
(2.61)E0(γ̂(t))<0∀t∈[-1,1].
Let S=V0∩∂B1Lp(Ω), where ∂B1Lp(Ω)={u∈Lp(Ω):∥u∥p=1}, and SC=S∩C01(Ω¯) be endowed with the topologies induced by V0 and C01(Ω¯), respectively. We set
(2.62)Γ0,C={γ∈C([-1,1],SC):γ(-1)=-φ1,γ(1)=φ1}.
Making use of the first inequality in assumption H(f) (iii), we fix numbers μ>λ2 and δ>0 such that (2.14) holds, and then let ρ0∈(0,μ-λ2). We recall the following variational expression for λ2 given by Cuesta et al. [32]:
(2.63)λ2=infγ∈Γ0maxu∈γ([-1,1])∥∇u∥pp,
where
(2.64)Γ0={γ∈C([-1,1],S):γ(-1)=-φ1,γ(1)=φ1}.
By (2.63) there exists γ∈Γ0 such that
(2.65)maxt∈[-1,1]∥∇γ(t)∥pp<λ2+ρ02.
Choose some number r with 0<r≤(λ2+ρ0)1/p-(λ2+ρ0/2)1/p. The density of SC in S implies that Γ0,C is dense in Γ0; so there is γ0∈Γ0,C satisfying
(2.66)maxt∈[-1,1]∥∇γ(t)-∇γ0(t)∥p<r.
Then the choice of r establishes
(2.67)maxt∈[-1,1]∥∇γ0(t)∥pp<λ2+ρ0.
The boundedness of the set γ0([-1,1])(Ω¯) in ℝensures the existence of some ε1>0 such that
(2.68)ε1|u(x)|≤δ∀x∈Ω∀u∈γ0([-1,1]).
Since u+,-u-∈int(C01(Ω¯)+) (see Theorem 2.4), for every u∈γ0([-1,1]) and any bounded neighborhood Vu of u in C01(Ω¯) there exist positive numbers hu and ju such that
(2.69)u+-1hv∈int(C01(Ω¯)+),-u-+1jv∈int(C01(Ω¯)+),
whenever h≥hu, j≥ju, and v∈Vu. This fact and the compactness of γ0([-1,1]) in C01(Ω¯) allow to determine a number ε0>0 for which one has
(2.70)u-(x)≤εu(x)≤u+(x)∀x∈Ω,u∈γ0([-1,1]),ε∈(0,ε0).
We now focus on the continuous path εγ0 in C01(Ω¯) joining -εφ1and εφ1 with a fixed constant ε satisfying 0<ε<min{ε0,ε1}. By (2.70), (2.67), (2.68), (2.14) with μ>λ2, and taking into account the choice of ρ0 as well as γ0([-1,1])⊂∂B1Lp(Ω) we obtain
(2.71)E0(εγ0(t))=εpp∥∇γ0(t)∥pp-∫Ω∫0εγ0(t)(x)f(x,τ0(x,s),λ)dsdx=εpp∥∇γ0(t)∥pp-∫Ω∫0εγ0(t)(x)f(x,s,λ)dsdx≤εpp(λ2+ρ0-μ)<0∀t∈[-1,1].
At this point we apply the second deformation lemma (see, e.g., [27, page 366]) to the C1 functional E+:V0→ℝ. Towards this let us denote(2.72)c+=c+(λ)=E+(εφ1),m+=m+(λ)=E+(u+),E+c+={u∈V0:E+(u)≤c+}.
It was already shown that u+ is the unique global minimizer of E+, and so we have m+<c+. Taking into account (2.55), E+ has no critical values in the interval (m+,c+] (for, otherwise, the minimality of the positive solution u+ of (1.4) would be contradicted). Using also that the functional E+ satisfies the Palais-Smale condition (because it is coercive), the second deformation lemma can be applied to E+yielding a continuous mapping η∈C([0,1]×E+c+,E+c+) such that η(0,u)=u and η(1,u)=u+ for all u∈E+c+, as well as E+(η(t,u))≤E+(u) whenever t∈[0,1] and u∈E+c+. Introducing γ+:[0,1]→V0 by
(2.73)γ+(t):=(η(t,εφ1))+:=max{η(t,εφ1),0}
for all t∈[0,1], it is seen that γ+ is a continuous path in V0 joining εφ1 and u+. (Note the mapping w↦w+ is continuous from V0 into itself.) The properties of the deformation η imply
(2.74)E0(γ+(t))=E+(γ+(t))≤E+(η(t,εφ1))=E+(εφ1)=E0(εφ1)<0
for all t∈[0,1]. Similarly, applying the second deformation lemma to the functional E-, we construct a continuous path γ-:[0,1]→V0 joining u- and -εφ1 such that
(2.75)E0(γ-(t))<0∀t∈[0,1].
The union of the curves γ-,  εγ0, and γ+ gives rise to a path γ̂∈Γ. We see from (2.75), (2.71), and (2.74) that (2.61) is satisfied. Hence (2.60) holds, and so u0≠0. Recalling that the critical points of E0 are in the order interval {u∈V0:u-(x)≤u(x)≤u+(x)a.e.x∈Ω} we derive that u0 is a nontrivial solution of (1.4) distinct from u- and u+, with u-≤u0≤u+. By the nonlinear regularity theory we have that u0∈C01(Ω¯). The extremality properties of the constant sign solutions u- and u+ as described in Theorem 2.4 force u0 to be sign-changing. This completes the proof.


### 2.4. Two Sign-Changing Solutions

The goal of this section is to show that under hypotheses stronger than those in Theorem2.5, problem (1.4) possesses at least two sign-changing solutions.The new hypotheses on the nonlinearityf(x,s,λ) in problem (1.4) are the following.H′(f)
f:Ω̅×ℝ×(0,λ̅)→ℝ, with λ̅>0, is a function such that f(x,0,λ)=0 for a.a. x∈Ω, whenever λ∈(0,λ̅).(i)
For allλ∈(0,λ̅), f(·,·,λ)∈C(Ω̅×ℝ).(ii)
There are constantsc>0, r∈(p-1,p*-1), and functions a(·,λ)∈L∞(Ω)+ (λ∈(0,λ̅)) with ∥a(·,λ)∥∞→0 as λ↓0 such that
(2.76)|f(x,s,λ)|≤a(x,λ)+c|s|rfora.a.x∈Ω∀(s,λ)∈ℝ×(0,λ̅).(iii)
For allλ∈(0,λ̅) there exist constants μ0=μ0(λ)>λ2, ν0=ν0(λ)>μ0 and a set Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.77)μ0<lim infs→0f(x,s,λ)|s|p-2s≤lim sups→0f(x,s,λ)|s|p-2s≤ν0
uniformly with respect to x∈Ωλ.(iv)
There exist constantsb-<0<b+ such that for all λ∈(0,λ̅) we have
(2.78)f(x,b-,λ)=0=f(x,b+,λ)∀x∈Ω,f(x,s,λ)<0∀x∈Ω,alls∈(b-,0),f(x,s,λ)>0∀x∈Ω,alls∈(0,b+).(v)
For everyλ∈(0,λ̅), there exist M=M(λ)>0 and μ=μ(λ)>p such that(2.79)0<μF(x,s,λ)≤f(x,s,λ)s∀x∈Ω,all|s|≥M.We notice that hypothesesH'(f) are stronger than H(f). In particular, for every λ∈(0,λ̅), we added the Ambrosetti-Rabinowitz condition for f(·,·,λ) (see hypothesis H'(f)(v)).We state now the main result of this section, which produces two sign-changing solutions for problem (1.4).Theorem 2.6.
Assume that hypothesesH'(f) are fulfilled. Then there exists a number λ0∈(0,λ̅) with the property that if λ∈(0,λ0), then problem (1.4) has a minimal (positive) solution u+=u+(λ)∈int(C01(Ω¯)+), a maximal (negative) solution u-=u-(λ)∈-int(C01(Ω¯)+), and two nontrivial sign-changing solutions u0=u0(λ),w0=w0(λ)∈C01(Ω¯) satisfying ∥u+∥∞<b, ∥u-∥∞<b, u-≤u0≤u+ a.e. in Ω (so ∥u0∥∞<b) and ∥w0∥∞≥b, where b:=min{b+,|b-|}.Proof.
Since hypothesesH'(f) are stronger than H(f), we can apply Theorem 2.5 with b=min{b+,|b-|}, which ensures the existence of a number λ0∈(0,λ̅) such that for every λ∈(0,λ0), problem (1.4) possesses a positive solution u+=u+(λ)∈int(C01(Ω¯)+), a negative solution u-=u-(λ)∈-int(C01(Ω¯)+), and a sign-changing solution u0=u0(λ)∈C01(Ω¯) with -b<u-≤u0≤u+<b. The proof of Theorem 2.5 shows that u+ and u- can be chosen to be the minimal positive solution and the maximal negative solution, respectively.
On the other hand, hypothesesH'(f) enable us to apply Theorem 1.1 of Bartsch et al. [33]. It follows that there exists a sign-changing solution w0=w0(λ)∈C01(Ω¯) (by the nonlinear regularity theory) with maxΩ¯w0≥b+ and minΩ¯w0≤b-. Therefore, we have ∥w0∥∞≥b, which shows that the sign-changing solutions u0 and w0 are different. This completes the proof.Remark 2.7.
In fact, under hypothesesH'(f), for λ∈(0,λ0), problem (1.4) admits at least six nontrivial solutions: two positive solutions, two negative solutions, and two sign-changing solutions, as seen in Theorem 5 in [34].


## 2.1. Hypotheses and Example

LetLq(Ω)+,1≤q≤+∞, denote the positive cone of Lq(Ω) given by (2.1)Lq(Ω)+={v∈Lq(Ω):v(x)≥0fora.a.x∈Ω}.
We impose the following hypotheses on the nonlinearity f(x,s,λ) in problem (1.4).H(f)
f:Ω×ℝ×(0,λ̅)→ℝ, with λ̅>0, is a function such that f(x,0,λ)=0 for a.a. x∈Ω, whenever λ∈(0,λ̅), and one has the following.(i)
For allλ∈(0,λ̅), f(·,·,λ) is Carathéodory (i.e., f(·,s,λ) is measurable for all s∈ℝ and f(x,·,λ) is continuous for almost all x∈Ω).(ii)
There are constantsc>0, r>p-1, and functions a(·,λ)∈L∞(Ω)+ (λ∈(0,λ̅)) with ∥a(·,λ)∥∞→0 as λ↓0 such that
(2.2)|f(x,s,λ)|≤a(x,λ)+c|s|rfora.a.x∈Ω∀(s,λ)∈ℝ×(0,λ̅).(iii)
For allλ∈(0,λ̅) there exist constants μ0=μ0(λ)>λ2, ν0=ν0(λ)>μ0 and a set Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.3)μ0<lim infs→0f(x,s,λ)|s|p-2s≤lim sups→0f(x,s,λ)|s|p-2s≤ν0
uniformly with respect to x∈Ωλ.InH(f)(iii), λ2 denotes the second eigenvalue of (-Δp,V0). As mentioned in the Introduction, the strengthening with respect to [26] (see also [25]) of the unilateral condition for the right-hand side f in (1.4), which enables us to obtain, in addition, sign-changing solutions, consists in adding the part involving the limit superior in H(f)(iii).Let us provide an example where all the assumptions formulated inH(f) are fulfilled.Example 2.1.
For the sake of simplicity we drop thex dependence for the function f in the right-hand side of (1.4). The function f:ℝ×(0,+∞)→ℝ given by
(2.4)f(s,λ)=λarctan(λ+λ2λ|s|p-2s)+c|s|r-1s∀(s,λ)∈ℝ×(0,+∞),
with c>0 and r>p-1, satisfies hypotheses H(f). Next we give an example of function f:ℝ×(0,+∞)→ℝ verifying assumptions H(f) which is generally not odd with respect to s:
(2.5)f(s,λ)={λarctan(a1λ+λ2λ|s|p-2s)+c1|s|r1-1sifs≤0,λarctan(a2λ+λ2λsp-1)+c2sr2ifs>0,
with λ>0, a1≥1, a2≥1, c1>0, c2>0, r1>p-1, r2>p-1.


## 2.2. Constant-Sign Solutions

The operator-Δp:V0→V0* is maximal monotone and coercive; therefore there exists a unique solution e∈V0 of the Dirichlet problem(2.6)e∈V0:-Δpe=1inV0*.
With s-=max{-s,0} for s∈ℝ, and using -e-∈V0 as a test function, we see that(2.7)∥∇e-∥pp=〈-Δpe,-e-〉=-∫Ωe-(x)dx≤0,
which implies that e≥0. From the nonlinear regularity theory (cf., e.g., [27, Theorem 1.5.6]) we have e∈C01(Ω¯). Then from the nonlinear strong maximum principle (see [28]) we infer that e∈int(C01(Ω¯)+). Here int(C01(Ω¯)+) denotes the interior of the positive cone C01(Ω¯)+={u∈C01(Ω¯):u(x)≥0,∀x∈Ω} in the Banach space C01(Ω¯)={u∈C1(Ω¯):u(x)=0,∀x∈∂Ω}, given by(2.8)int(C01(Ω¯)+)={u∈C01(Ω¯):u(x)>0,∀x∈Ω,and∂u∂n(x)<0,∀x∈∂Ω},
where n=n(x) is the outer unit normal at x∈∂Ω.Lemma 2.2.
Let the datac, r, and a(·,λ) be as in H(f)(ii). Then for every constant θ>0 there is λ0∈(0,λ̅) with the property that if λ∈(0,λ0), one can choose ξ0=ξ0(λ)∈(0,θ) such that
(2.9)c(ξ0∥e∥∞)r+∥a(·,λ)∥∞<ξ0p-1.Proof.
On the contrary there would exist a constantθ>0 and a sequence λn↓0 as n→∞ such that
(2.10)c(ξ∥e∥∞)r+∥a(·,λn)∥∞≥ξp-1∀n∈ℕ,ξ∈(0,θ).
Letting n→∞ we get c∥e∥∞rξr-p+1≥1 for all ξ∈(0,θ) because we have ∥a(·,λ)∥∞→0 as λ↓0. Since r>p-1, a contradiction is achieved as ξ↓0. Therefore (2.9) holds true.We denote byλ1 the first eigenvalue of (-Δp,V0) and by φ1 the eigenfunction of (-Δp,V0) corresponding to λ1 satisfying (2.11)φ1∈int(C01(Ω¯)+),∥φ1∥p=1.Lemma 2.3.
AssumeH(f)(i) and (ii) and the following weaker form of hypothesis H(f)(iii): for all λ∈(0,λ̅) there exist μ0=μ0(λ)>λ1 and Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.12)μ0<lim infs→0f(x,s,λ)|s|p-2s
uniformly with respect to x∈Ωλ.
Fix a constantθ>0 and consider the corresponding number λ0∈(0,λ̅) obtained in Lemma 2.2. Then for any λ∈(0,λ0) the function u¯=ξ0e∈int(C01(Ω¯)+), with ξ0∈(0,θ) given by Lemma 2.2, is a supersolution for problem (1.4), and the function u̲=εφ1∈int(C01(Ω¯)+) is a subsolution of problem (1.4) provided that the number ε>0 is sufficiently small.Proof.
For a fixedλ∈(0,λ0), from (2.9) and H(f)(ii) we derive
(2.13)-Δpu¯=ξ0p-1>∥a(·,λ)∥∞+c∥u¯∥∞r≥f(·,u¯(·),λ),
which says that u¯=ξ0e is a supersolution for problem (1.4).
On the other hand, by hypothesis we can findμ=μ(λ)>λ1 and δ=δ(λ)>0 such that(2.14)μ<f(x,s,λ)|s|p-2sfora.a.x∈Ω∀0<|s|≤δ.
Choose ε∈(0,δ/∥φ1∥∞). Then by (2.14) we have
(2.15)-Δp(εφ1)=λ1εp-1φ1p-1<μεp-1φ1p-1<f(x,εφ1(x),λ)fora.a.x∈Ω,
which ensures that u̲=εφ1 is a subsolution of problem (1.4).The following result which asserts the existence of two solutions of problem (1.4) having opposite constant sign and being extremal plays an important role in the proof of the existence of sign-changing solutions.Theorem 2.4.
AssumeH(f)(i) and (ii) and the following weaker form of H(f)(iii): for all λ∈(0,λ̅) there exist constants μ0=μ0(λ)>λ1, ν0=ν0(λ)>μ0 and a set Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.16)μ0<lim infs→0f(x,s,λ)|s|p-2s≤lim sups→0f(x,s,λ)|s|p-2s≤ν0
uniformly with respect to x∈Ωλ. Then for all b>0 there exists a number λ0∈(0,λ̅) with the property that if λ∈(0,λ0), then there is a constant ξ0=ξ0(λ)∈(0,b/∥e∥∞) such that problem (1.4) has a least positive solution u+=u+(λ)∈int(C01(Ω¯)+) in the order interval [0,ξ0e] and a greatest negative solution u-=u-(λ)∈-int(C01(Ω¯)+) in the order interval [-ξ0e,0].Proof.
Since the proof of the existence of the greatest negative solution follows the same lines, we only provide the arguments for the existence of the least positive solution.
Applying Lemma2.3 for θ=b/∥e∥∞we find λ0∈(0,λ̅) as therein. Fix λ∈(0,λ0). Lemma 2.3 ensures that u¯=ξ0e∈int(C01(Ω¯)+) is a supersolution for problem (1.4), with ξ0∈(0,b/∥e∥∞) given by Lemma 2.2, and u̲=εφ1∈int(C01(Ω¯)+) is a subsolution for problem (1.4) if ε>0 is small enough. Passing eventually to a smaller ε>0, we may assume that εφ1≤ξ0e. Then by the method of sub-supersolution we know that in the order interval [εφ1,ξ0e] there is a least (i.e., smallest) solution uε=uε(λ)∈int(C01(Ω¯)+) of problem (1.4) (see [29]).
We thus obtain that for every positive integern sufficiently large there is a least solution un∈int(C01(Ω¯)+) of problem (1.4) in the order interval [(1/n)φ1,ξ0e]. Clearly, we have(2.17)un↓u+pointwise,
with some function u+:Ω→ℝ satisfying 0≤u+≤ξ0e. First we claim that
(2.18)u+isasolutionofproblem(1.4).
Taking into account that un solves (1.4), and the fact that un belongs to the order interval [0,ξ0e], from H(f)(ii) we see that
(2.19)∥∇un∥pp=∫Ωf(x,un(x),λ)un(x)dx≤∫Ω(a(x,λ)+cξ0re(x)r)ξ0e(x)dx,
which implies the boundedness of the sequence (un) in V0. Then due to (2.17) we have that u+∈V0 as well as
(2.20)un⇀u+inV0,un→u+inLp(Ω)anda.e.inΩ.
Since un solves problem (1.4), one has
(2.21)〈-Δpun,φ〉=∫Ωf(x,un(x),λ)φ(x)dx,∀φ∈V0.
Setting φ=un-u+ in (2.21) gives
(2.22)〈-Δpun,un-u+〉=∫Ωf(x,un(x),λ)(un(x)-u+(x))dx.
As already noticed that the sequence (f(·,un(·),λ) is uniformly bounded on Ω, so (2.20) and (2.22) yield
(2.23)limn→∞〈-Δpun,un-u+〉=0.
The S+-property of -Δp on V0 implies
(2.24)un→u+inV0asn→∞.
The strong convergence in (2.24) and Lebesgue's dominated convergence theorem permit to pass to the limit in (2.21) that results in (2.18).
By (2.18) and the nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) it turns out u+∈C01(Ω¯). The choice of ξ0 guarantees that(2.25)0≤u+(x)≤ξ0e(x)≤bfora.e.x∈Ω.
Thus, from (2.18), assumptions H(f)(ii) and (iii), and the boundedness of u+, we get
(2.26)-Δpu+(x)=f(x,u+(x),λ)≥-ĉu+(x)p-1fora.a.x∈Ω,
with a constant ĉ>0. Applying the nonlinear strong maximum principle (cf. [28]) we conclude that either u+=0 or u+∈int(C01(Ω¯)+).
We claim that(2.27)u+∈int(C01(Ω¯)+).
Assume on the contrary that u+=0. Then (2.17) becomes
(2.28)un(x)↓0∀x∈Ω.
Since un≥(1/n)φ1, we may consider
(2.29)ũn=un∥∇un∥p∀n.
Along a relabelled subsequence we may suppose
(2.30)ũn⇀ũinV0,ũn→ũinLp(Ω)anda.e.inΩ
for some ũ∈V0. Moreover, one can find a function w∈Lp(Ω)+ such that |ũn(x)|≤w(x) for almost all x∈Ω. Relation (2.21) reads
(2.31)〈-Δpũn,φ〉=∫Ωf(x,un(x),λ)∥∇un∥pp-1φdx,∀φ∈V0.
Setting φ=ũn-ũ leads to
(2.32)〈-Δpũn,ũn-ũ〉=∫Ωf(x,un(x),λ)∥∇un∥pp-1(ũn-ũ)dx.
By H(f)(iii) we know that there exist constants c0=c0(λ)>λ1 and α=α(λ)>0 such that
(2.33)|f(x,s,λ)|≤c0|s|p-1fora.a.x∈Ω,∀|s|<α,
while H(f)(ii) entails
(2.34)|f(x,s,λ)|≤a(x,λ)+c|s|r≤(∥a(·,λ)∥∞αr+c)|s|r
for a.a. x∈Ω and for all |s|≥α. Combining the two estimates gives
(2.35)|f(x,s,λ)|≤c0|s|p-1+c1|s|rfora.a.x∈Ω,∀s∈ℝ
with a constant c1=c1(λ)>0. Since un∈[(1/n)φ1,ξ0e], r>p-1 and (2.35) holds, there exists a constant C>0 such that
(2.36)|f(x,un(x),λ)|un(x)p-1≤Cfora.a.x∈Ω,∀n.
We see from (2.36) that
(2.37)|f(x,un(x),λ)|∥∇un∥pp-1|ũn(x)-ũ(x)|≤Cw(x)p-1(w(x)+|ũ(x)|)fora.a.x∈Ω.
Then, because the right-hand side of the above inequality is in L1(Ω), by means of (2.30) and (2.36) we can apply Lebesgue's dominated convergence theorem to get
(2.38)limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1(ũn-ũ)dx=0.
Consequently, from (2.32) we obtain
(2.39)limn→∞〈-Δpũn,ũn-ũ〉=0.
The S+-property of -Δp on V0 implies
(2.40)ũn→ũinV0asn→∞.
On the basis of (2.31) and (2.40) it follows
(2.41)〈-Δpũ,φ〉=limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1φdx,∀φ∈V0.
Notice from (2.36) that
(2.42)|f(x,un(x),λ)|∥∇un∥pp-1|φ(x)|≤Cw(x)p-1|φ(x)|
for a.a. x∈Ω and for all φ∈V0. We are thus allowed to apply Fatou's lemma which in conjunction with (2.28), (2.30), and (2.16) ensures
(2.43)limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1φ(x)dx=limn→∞∫Ωf(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x)dx≥∫Ωlim infn→∞(f(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x))dx≥μ0∫Ωũ(x)p-1φ(x)dx
for all φ∈V0,+:=V0∩Lp(Ω)+. Thanks to (2.41) we obtain
(2.44)〈-Δpũ,φ〉≥μ0∫Ωũ(x)p-1φ(x)dx,∀φ∈V0,+.
Owing to (2.42) we may once again use Fatou's lemma; so according to (2.28), (2.30), and the last part of (2.16), we find
(2.45)limn→∞∫Ωf(x,un(x),λ)∥∇un∥pp-1φ(x)dx=limn→∞∫Ωf(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x)dx≤∫Ωlim supn→∞(f(x,un(x),λ)un(x)p-1ũn(x)p-1φ(x))dx≤ν0∫Ωũ(x)p-1φ(x)dx
for all φ∈V0,+. Then (2.41) ensures
(2.46)〈-Δpũ,φ〉≤ν0∫Ωũ(x)p-1φ(x)dx,∀φ∈V0,+.
Combining (2.44) and (2.46) results in
(2.47)μ0ũp-1≤-Δpũ≤ν0ũp-1a.e.inΩ,
which guarantees to have ũ∈L∞(Ω) (see [27, Theorem 1.5.5]). Since by (2.47) we know that Δpũ∈L∞(Ω), we are in a position to address Theorem 1.5.6 in [27], which provides ũ∈C1,β(Ω¯) with some β∈(0,1). This regularity up to the boundary and the fact that ũ≥0 a.e. in Ω and (2.47) enable us to refer to the strong maximum principle (see Theorem 5 of Vázquez [28]). Recalling that ũ does not vanish identically on Ω (because ∥∇ũ∥p=1) we deduce that ũ(x)>0 for all x∈Ω and (∂ũ/∂n)(x)<0 for all x∈∂Ω which amounts to saying ũ∈int(C01(Ω¯)+). Consequently, there exist constants k0>0 and k1>0 such that
(2.48)k0φ1≤ũ<k1φ1a.e.inΩ.
Following [30] let us denote
(2.49)I(u,v)=〈-Δpu,up-vpup-1〉-〈-Δpv,up-vpvp-1〉
whenever (u,v)∈DI, where
(2.50)DI={(w1,w2)∈(V0)2:wi≥0,wiwj∈L∞(Ω)fori,j∈{1,2}}.
Relation (2.48) justifies that (k1φ1,ũ)∈DI. Then Proposition 1 of Anane [30] implies I(k1φ1,ũ)≥0. On the other hand a direct computation based on (2.48) and (2.47) shows
(2.51)I(k1φ1,ũ)=〈-Δp(k1φ1),(k1φ1)p-ũp(k1φ1)p-1〉-〈-Δpũ,(k1φ1)p-ũpũp-1〉≤(λ1-μ0)∫Ω((k1φ1)p-ũp)dx<0.
This contradiction proves that the claim in (2.27) holds true.
In view of (2.18) it remains to establish that u+ is the smallest positive solution of problem (1.4) in the interval [0,u¯]. Let u∈V0 be a positive solution to (1.4) in [0,u¯]. Since u∈L∞(Ω), then (1.4) and H(f)(ii) allow to deduce that -Δpu∈L∞(Ω). Using Theorem 1.5.6 of [27] leads to u∈C01(Ω¯). Then, as u is a solution to (1.4) and u∈[0,u¯], with ∥u¯∥∞<b, by means of hypotheses H(f)(ii) and (iii), we are able to apply the strong maximum principle. So we get u∈int(C01(Ω¯)+), hence u∈[(1/n)φ1,u¯] for n sufficiently large. The fact that un is the least solution of (1.4) in [(1/n)φ1,u¯] ensures un≤u. Taking into account (2.17), we obtain u+≤u. This completes the proof.


## 2.3. Sign-Changing Solution

The main result of this section is as follows.Theorem 2.5.
Under hypothesesH(f), for all b>0, there exists a number λ0∈(0,λ̅) with the property that if λ∈(0,λ0), then problem (1.4) has a (positive) solution u+=u+(λ)∈int(C01(Ω¯)+), a (negative) solution u-=u-(λ)∈-int(C01(Ω¯)+), and a nontrivial sign-changing solution u0=u0(λ)∈C01(Ω¯) satisfying ∥u+∥∞<b, ∥u-∥∞<b, ∥u0∥∞<b.Proof.
Letb>0. Consider the positive number λ0 given by Theorem 2.4 and fix λ∈(0,λ0). Let u+∈int(C01(Ω¯)+) and u-∈-int(C01(Ω¯)+) be the two extremal solutions determined in Theorem 2.4. We introduce on Ω×ℝ the truncation functions
(2.52)τ+(x,s)={0ifs≤0,sif0<s<u+(x),u+(x)ifs≥u+(x),τ-(x,s)={u-(x)ifs≤u-(x),sifu-(x)<s<0,0ifs≥0,τ0(x,s)={u-(x)ifs≤u-(x),sifu-(x)<s<u+(x),u+(x)ifs≥u+(x)
and then define the following associated functionals:
(2.53)E+(u)=1p∥∇u∥pp-∫Ω∫0u(x)f(x,τ+(x,s),λ)dsdx,∀u∈V0,E-(u)=1p∥∇u∥pp-∫Ω∫0u(x)f(x,τ-(x,s),λ)dsdx,∀u∈V0,E0(u)=1p∥∇u∥pp-∫Ω∫0u(x)f(x,τ0(x,s),λ)dsdx,∀u∈V0.
It is clear that E+,E-,E0∈C1(V0).
We observe that ifv is a critical point of E+, then(2.54)〈-Δpv+Δpu+,(v-u+)+〉=∫Ω(f(x,τ+(x,v(x)),λ)-f(x,u+(x),λ))(v-u+)+dx=0
which implies v≤u+. Similarly, it follows that v≥0. This leads to
(2.55)visacriticalpointofE+⇒0≤v(x)≤u+(x)fora.a.x∈Ω.
Since the functionE+ is coercive and weakly lower semicontinuous, there exists a global minimizer z+∈V0 of it. Using (2.14), it is seen that(2.56)E+(z+)=infV0E+<0,
and so z+≠0. Relation (2.55) shows that z+ is a nontrivial solution of problem (1.4) belonging to the order interval [0,u+]. Via assumptions H(f)(ii) and (iii) and the boundedness of z+, we may apply the strong maximum principle which ensures z+>0 on Ω. In view of the minimality property of u+ as stated in Theorem 2.4, it follows that z+=u+. In fact, u+ is the unique global minimizer of E+.
Sinceu+∈int(C01(Ω¯)+), there exists a neighborhood 𝒰 of u+ in the space C01(Ω¯) such that 𝒰⊂C01(Ω¯)+. Therefore E0=E+ on 𝒰, which guarantees that u+ is a local minimizer of E0 on C01(Ω¯). It results that u+ is also a local minimizer of E0 on the space V0 (see [27], pages 655-656 ). Employing the functional E- and proceeding as in the case of u+, we establish that u- is a local minimizer of E0 on V0.
As in the case of (2.55), we verify that every critical point of E0 belongs to the set {u∈V0:u-(x)≤u(x)≤u+(x)a.e.x∈Ω}, which implies that every critical point of E0 is a solution to problem (1.4). The functional E0 is coercive, weakly lower semicontinuous, with infV0E0<0. Thus E0 has a global minimizer y0∈V0 with y0≠0. The above properties ensure that y0 is a nontrivial solution of problem (1.4) belonging to the order interval [u-,u+]. Assume y0≠u+ and y0≠u-. We claim that y0 changes sign. Indeed, if not, y0 would have constant sign, for instance y0≥0 a.e. on Ω. Using assumptions H(f)(ii) and (iii) and the boundedness of y0, we may apply the strong maximum principle which leads to y0>0 on Ω. This is impossible because it contradicts the minimality property of the solution u+ as given by Theorem 2.4. According to the claim, we obtain the conclusion of the theorem setting u0=y0.
Thus, the proof reduces to consider the casesy0=u+ or y0=u-. To make a choice, suppose y0=u+. We may also admit that u- is a strict local minimizer of E0. This is true since on the contrary we would find (infinitely many) critical points x0 of E0 belonging to the order interval [u-,u+] which are different from 0, u-, u+, and if x0 does not change sign, taking into account the strong maximum principle, the extremality properties of the solutions u-, u+ given in Theorem 2.4 will be contradicted. A straightforward argument allows then to find ρ∈(0,∥u+-u-∥) such that(2.57)E0(u+)≤E0(u-)<inf{E0(u):u∈∂Bρ(u-)},
where ∂Bρ(u-)={u∈V0:∥u-u-∥=ρ}. Relation (2.57) in conjunction with the Palais-Smale condition (which holds for E0 due to its coercivity) enables us to apply the mountain pass theorem to the functional E0 (see, e.g., [31]). In this way we get u0∈V0 satisfying E′0(u0)=0 and
(2.58)inf{E0(u):u∈∂Bρ(u-)}≤E0(u0)=infγ∈Γmaxt∈[-1,1]E0(γ(t)),
where
(2.59)Γ={γ∈C([-1,1],V0):γ(-1)=u-,γ(1)=u+}.
We infer from (2.57) and (2.58) that u0≠u-and u0≠u+.
The next step in the proof is to show that(2.60)E0(u0)<0.
By the equality in (2.58), it suffices to produce a path γ̂∈Γ such that
(2.61)E0(γ̂(t))<0∀t∈[-1,1].
Let S=V0∩∂B1Lp(Ω), where ∂B1Lp(Ω)={u∈Lp(Ω):∥u∥p=1}, and SC=S∩C01(Ω¯) be endowed with the topologies induced by V0 and C01(Ω¯), respectively. We set
(2.62)Γ0,C={γ∈C([-1,1],SC):γ(-1)=-φ1,γ(1)=φ1}.
Making use of the first inequality in assumption H(f) (iii), we fix numbers μ>λ2 and δ>0 such that (2.14) holds, and then let ρ0∈(0,μ-λ2). We recall the following variational expression for λ2 given by Cuesta et al. [32]:
(2.63)λ2=infγ∈Γ0maxu∈γ([-1,1])∥∇u∥pp,
where
(2.64)Γ0={γ∈C([-1,1],S):γ(-1)=-φ1,γ(1)=φ1}.
By (2.63) there exists γ∈Γ0 such that
(2.65)maxt∈[-1,1]∥∇γ(t)∥pp<λ2+ρ02.
Choose some number r with 0<r≤(λ2+ρ0)1/p-(λ2+ρ0/2)1/p. The density of SC in S implies that Γ0,C is dense in Γ0; so there is γ0∈Γ0,C satisfying
(2.66)maxt∈[-1,1]∥∇γ(t)-∇γ0(t)∥p<r.
Then the choice of r establishes
(2.67)maxt∈[-1,1]∥∇γ0(t)∥pp<λ2+ρ0.
The boundedness of the set γ0([-1,1])(Ω¯) in ℝensures the existence of some ε1>0 such that
(2.68)ε1|u(x)|≤δ∀x∈Ω∀u∈γ0([-1,1]).
Since u+,-u-∈int(C01(Ω¯)+) (see Theorem 2.4), for every u∈γ0([-1,1]) and any bounded neighborhood Vu of u in C01(Ω¯) there exist positive numbers hu and ju such that
(2.69)u+-1hv∈int(C01(Ω¯)+),-u-+1jv∈int(C01(Ω¯)+),
whenever h≥hu, j≥ju, and v∈Vu. This fact and the compactness of γ0([-1,1]) in C01(Ω¯) allow to determine a number ε0>0 for which one has
(2.70)u-(x)≤εu(x)≤u+(x)∀x∈Ω,u∈γ0([-1,1]),ε∈(0,ε0).
We now focus on the continuous path εγ0 in C01(Ω¯) joining -εφ1and εφ1 with a fixed constant ε satisfying 0<ε<min{ε0,ε1}. By (2.70), (2.67), (2.68), (2.14) with μ>λ2, and taking into account the choice of ρ0 as well as γ0([-1,1])⊂∂B1Lp(Ω) we obtain
(2.71)E0(εγ0(t))=εpp∥∇γ0(t)∥pp-∫Ω∫0εγ0(t)(x)f(x,τ0(x,s),λ)dsdx=εpp∥∇γ0(t)∥pp-∫Ω∫0εγ0(t)(x)f(x,s,λ)dsdx≤εpp(λ2+ρ0-μ)<0∀t∈[-1,1].
At this point we apply the second deformation lemma (see, e.g., [27, page 366]) to the C1 functional E+:V0→ℝ. Towards this let us denote(2.72)c+=c+(λ)=E+(εφ1),m+=m+(λ)=E+(u+),E+c+={u∈V0:E+(u)≤c+}.
It was already shown that u+ is the unique global minimizer of E+, and so we have m+<c+. Taking into account (2.55), E+ has no critical values in the interval (m+,c+] (for, otherwise, the minimality of the positive solution u+ of (1.4) would be contradicted). Using also that the functional E+ satisfies the Palais-Smale condition (because it is coercive), the second deformation lemma can be applied to E+yielding a continuous mapping η∈C([0,1]×E+c+,E+c+) such that η(0,u)=u and η(1,u)=u+ for all u∈E+c+, as well as E+(η(t,u))≤E+(u) whenever t∈[0,1] and u∈E+c+. Introducing γ+:[0,1]→V0 by
(2.73)γ+(t):=(η(t,εφ1))+:=max{η(t,εφ1),0}
for all t∈[0,1], it is seen that γ+ is a continuous path in V0 joining εφ1 and u+. (Note the mapping w↦w+ is continuous from V0 into itself.) The properties of the deformation η imply
(2.74)E0(γ+(t))=E+(γ+(t))≤E+(η(t,εφ1))=E+(εφ1)=E0(εφ1)<0
for all t∈[0,1]. Similarly, applying the second deformation lemma to the functional E-, we construct a continuous path γ-:[0,1]→V0 joining u- and -εφ1 such that
(2.75)E0(γ-(t))<0∀t∈[0,1].
The union of the curves γ-,  εγ0, and γ+ gives rise to a path γ̂∈Γ. We see from (2.75), (2.71), and (2.74) that (2.61) is satisfied. Hence (2.60) holds, and so u0≠0. Recalling that the critical points of E0 are in the order interval {u∈V0:u-(x)≤u(x)≤u+(x)a.e.x∈Ω} we derive that u0 is a nontrivial solution of (1.4) distinct from u- and u+, with u-≤u0≤u+. By the nonlinear regularity theory we have that u0∈C01(Ω¯). The extremality properties of the constant sign solutions u- and u+ as described in Theorem 2.4 force u0 to be sign-changing. This completes the proof.


## 2.4. Two Sign-Changing Solutions

The goal of this section is to show that under hypotheses stronger than those in Theorem2.5, problem (1.4) possesses at least two sign-changing solutions.The new hypotheses on the nonlinearityf(x,s,λ) in problem (1.4) are the following.H′(f)
f:Ω̅×ℝ×(0,λ̅)→ℝ, with λ̅>0, is a function such that f(x,0,λ)=0 for a.a. x∈Ω, whenever λ∈(0,λ̅).(i)
For allλ∈(0,λ̅), f(·,·,λ)∈C(Ω̅×ℝ).(ii)
There are constantsc>0, r∈(p-1,p*-1), and functions a(·,λ)∈L∞(Ω)+ (λ∈(0,λ̅)) with ∥a(·,λ)∥∞→0 as λ↓0 such that
(2.76)|f(x,s,λ)|≤a(x,λ)+c|s|rfora.a.x∈Ω∀(s,λ)∈ℝ×(0,λ̅).(iii)
For allλ∈(0,λ̅) there exist constants μ0=μ0(λ)>λ2, ν0=ν0(λ)>μ0 and a set Ωλ⊂Ω with Ω∖Ωλ of Lebesgue measure zero such that
(2.77)μ0<lim infs→0f(x,s,λ)|s|p-2s≤lim sups→0f(x,s,λ)|s|p-2s≤ν0
uniformly with respect to x∈Ωλ.(iv)
There exist constantsb-<0<b+ such that for all λ∈(0,λ̅) we have
(2.78)f(x,b-,λ)=0=f(x,b+,λ)∀x∈Ω,f(x,s,λ)<0∀x∈Ω,alls∈(b-,0),f(x,s,λ)>0∀x∈Ω,alls∈(0,b+).(v)
For everyλ∈(0,λ̅), there exist M=M(λ)>0 and μ=μ(λ)>p such that(2.79)0<μF(x,s,λ)≤f(x,s,λ)s∀x∈Ω,all|s|≥M.We notice that hypothesesH'(f) are stronger than H(f). In particular, for every λ∈(0,λ̅), we added the Ambrosetti-Rabinowitz condition for f(·,·,λ) (see hypothesis H'(f)(v)).We state now the main result of this section, which produces two sign-changing solutions for problem (1.4).Theorem 2.6.
Assume that hypothesesH'(f) are fulfilled. Then there exists a number λ0∈(0,λ̅) with the property that if λ∈(0,λ0), then problem (1.4) has a minimal (positive) solution u+=u+(λ)∈int(C01(Ω¯)+), a maximal (negative) solution u-=u-(λ)∈-int(C01(Ω¯)+), and two nontrivial sign-changing solutions u0=u0(λ),w0=w0(λ)∈C01(Ω¯) satisfying ∥u+∥∞<b, ∥u-∥∞<b, u-≤u0≤u+ a.e. in Ω (so ∥u0∥∞<b) and ∥w0∥∞≥b, where b:=min{b+,|b-|}.Proof.
Since hypothesesH'(f) are stronger than H(f), we can apply Theorem 2.5 with b=min{b+,|b-|}, which ensures the existence of a number λ0∈(0,λ̅) such that for every λ∈(0,λ0), problem (1.4) possesses a positive solution u+=u+(λ)∈int(C01(Ω¯)+), a negative solution u-=u-(λ)∈-int(C01(Ω¯)+), and a sign-changing solution u0=u0(λ)∈C01(Ω¯) with -b<u-≤u0≤u+<b. The proof of Theorem 2.5 shows that u+ and u- can be chosen to be the minimal positive solution and the maximal negative solution, respectively.
On the other hand, hypothesesH'(f) enable us to apply Theorem 1.1 of Bartsch et al. [33]. It follows that there exists a sign-changing solution w0=w0(λ)∈C01(Ω¯) (by the nonlinear regularity theory) with maxΩ¯w0≥b+ and minΩ¯w0≤b-. Therefore, we have ∥w0∥∞≥b, which shows that the sign-changing solutions u0 and w0 are different. This completes the proof.Remark 2.7.
In fact, under hypothesesH'(f), for λ∈(0,λ0), problem (1.4) admits at least six nontrivial solutions: two positive solutions, two negative solutions, and two sign-changing solutions, as seen in Theorem 5 in [34].


## 3. Problem (1.7) for Parameters a and b being Large

The main goal of this section is to provide a detailed multiplicity analysis of the nonsmooth elliptic problem (1.7) in dependence of the two parameters a and b. Conditions in terms of the Fučik spectrum are formulated that ensure the existence of sign-changing solutions. As for the precise formulation of this result we recall the Fučik spectrum, see, for example, [13].The setΣp of those points (μ1,μ2)∈ℝ2 for which the problem(3.1)u∈V0:-Δpu=μ1(u+)p-1-μ2(u-)p-1inV0*
has a nontrivial solution is called the Fučik spectrum of the negative p-Laplacian on Ω. Hence, Σp clearly contains the two lines λ1×ℝ and ℝ×λ1 with λ1 being the first Dirichlet eigenvalue of -Δp. In addition, the spectrum σ(-Δp) of the negative p-Laplacian has an unbounded sequence of variational eigenvalues (λl), l∈ℕ, satisfying a standard min-max characterization, and Σp contains the corresponding sequence of points (λl,λl), l∈ℕ. A first nontrivial curve 𝒞 in Σp through (λ2,λ2) asymptotic to λ1×ℝ and ℝ×λ1 at infinity was constructed and variationally characterized by a mountain-pass procedure by Cuesta et al. [32] (see Figure 1), which implies the existence of a continuous path in {u∈V0:I(a,b)(u)<0,∥u∥p=1} joining -φ1 and φ1 provided (a,b) is above the curve 𝒞. Here the functional I(a,b) on V0 is given by (3.2)I(a,b)(u)=∫Ω(|∇u|p-a(u+)p-b(u-)p)dx.
The hypothesis on the parameters a and b that will finally ensure the existence of sign-changing solutions is as follows. (H)
Let(a,b)∈ℝ+2 be above the curve 𝒞 of the Fučik spectrum constructed in [32]; see Figure 1.Figure 1
Fučik Spectrum.


### 3.1. Hypotheses, Definitions, and Preliminaries

We impose the following hypotheses on the nonlinearityg:Ω×ℝ→ℝ whose primitive is G of problem (1.7)(g1)
(x,s)↦g(x,s) is measurable in each variable separately.(g2)
There existsc>0, and q∈[p,p*) such that
(3.3)|g(x,s)|≤c(1+|s|q-1)
for a.a. x∈Ω and for all s∈ℝ, where p* denotes the critical Sobolev exponent which is p*=Np/(N-p) if p<N, and p*=+∞ if p≥N.(g3)
One has(3.4)lims→0g(x,s)|s|p-2s=0uniformlywithrespecttoa.a.x∈Ω.(g4)
One has(3.5)lim|s|→∞g(x,s)|s|p-2s=+∞uniformlywithrespecttoa.a.x∈Ω.In view of assumptions (g1) and (g2) the functions↦G(x,s) is locally Lipschitz and the functional 𝒢:Lq(Ω)→ℝ defined by(3.6)𝒢(u):=∫ΩG(x,u(x))dx,u∈Lq(Ω)
is well defined and locally Lipschitz continuous as well. The generalized gradients ∂G(x,·) and ∂𝒢 can be characterized as follows: Define for every (x,t)∈Ω×ℝ, (3.7)g1(x,t):=limδ→0+essinf|τ-t|<δg(x,τ),g2(x,t):=limδ→0+esssup|τ-t|<δg(x,τ).
Proposition 1.7 in [35] ensures that (3.8)∂G(x,ξ)=[g1(x,ξ),g2(x,ξ)],
while Theorem 4.5.19 of [36] implies (3.9)∂𝒢(u)⊆{w∈Lq′(Ω):g1(x,u(x))≤w(x)≤g2(x,u(x))a.e.inΩ}
with q′:=q/(q-1). The next result is an immediate consequence of [37, Proposition 2.1.5].Lemma 3.1.
Supposeun→u in V0, wn⇀w in Lq′(Ω), and wn∈∂𝒢(un) for all n∈ℕ. Then w∈∂𝒢(u).Definition 3.2.
A functionu∈V0 is called a solution of (1.7) if there is an η∈Lq′(Ω) such that
(3.10)η(x)∈∂G(x,u(x))fora.a.x∈Ω,∫Ω|∇u|p-2∇u∇φdx+∫Ω(η-a(u+)p-1+b(u-)p-1)φdx=0,∀φ∈V0.Remark 3.3.
Due to assumption (g3) we haveg1(x,0)≤0≤g2(x,0) for almost all x∈Ω. Hence, in view of (3.8), problem (1.7) always possesses the trivial solution.Definition 3.4.
A functionu̲∈V:=W1,p(Ω) is called a subsolution of (1.7) if u̲|∂Ω≤0, and if there is an η̲∈Lq′(Ω) such that
(3.11)η̲(x)∈∂G(x,u̲(x))fora.a.x∈Ω,∫Ω|∇u̲|p-2∇u̲∇φdx+∫Ω(η̲-a(u̲+)p-1+b(u̲-)p-1)φdx≤0,∀φ∈V0∩Lp(Ω)+.Similarly, we define a supersolution as follows.Definition 3.5.
A functionu¯∈V is called a supersolution of (1.7) if u¯|∂Ω≥0, and if there is an η¯∈Lq′(Ω) such that
(3.12)η¯(x)∈∂G(x,u¯(x))fora.a.x∈Ω,∫Ω|∇u¯|p-2∇u¯∇φdx+∫Ω(η¯-a(u¯+)p-1+b(u¯-)p-1)φdx≥0,∀φ∈V0∩Lp(Ω)+.Lemma 3.6.
Lete be the uniquely defined solution of (2.6). If a>λ1, then there exists a constant αa>0 such that for any b∈ℝ the function αae is a positive supersolution of problem (1.7).Proof.
Leta>λ1. By (g4) there is sa>0 such that
(3.13)g(x,s)sp-1>afora.a.x∈Ω∀s>sa,
and by (g2) we get
(3.14)|g(x,s)-asp-1|≤|g(x,s)|+asp-1≤ca,fora.a.x∈Ω∀s∈[0,sa],
which implies
(3.15)g(x,s)≥asp-1-cafora.a.x∈Ω∀s≥0,
and thus in view of the definition of g2 we obtain
(3.16)g2(x,s)≥asp-1-cafora.a.x∈Ω∀s≥0,
Let u̅=αae, where αa is a positive constant to be specified. Then we get
(3.17)-Δpu̅-a(u̅+)p-1+b(u̅-)p-1+g2(x,u̅)=αap-1-aαap-1ep-1+g2(x,αae)≥αap-1-ca,
which shows that for αa:=ca1/(p-1) the function αae is in fact a supersolution of (1.7) with η¯(x)=g2(x,u̅(x)).In a similar way the following lemma on the existence of a negative subsolution can be proved.Lemma 3.7.
Lete be the uniquely defined solution of (2.6). If b>λ1, then there exists a constant βb>0 such that for any a∈ℝ the function -βbe is a negative subsolution of problem (1.7).In the next lemma we demonstrate that small constant multiples ofφ1 may be sub- and supersolutions of (1.7). More precisely we have the following result.Lemma 3.8.
Letφ1 be the normalized positive eigenfunction corresponding to the first eigenvalue λ1 of (-Δp,V0). If a>λ1, then for ε>0 sufficiently small and any b∈ℝ the function εφ1 is a positive subsolution of problem (1.7). If b>λ1, then for ε>0 sufficiently small and any a∈ℝ the function -εφ1 is a negative supersolution of problem (1.7).Proof.
By (g3) there is a constantδa>0 such that
(3.18)|g(x,s)||s|p-1<a-λ1fora.a.x∈Ω∀0<|s|≤δa,
which implies
(3.19)|g(x,s)|≤(a-λ1)sp-1fora.a.x∈Ω,∀s:0≤s≤δa.
Define u̲=εφ1 with ε>0. Applying (3.19) and the definition of g1 we get
(3.20)-Δpu̲-a(u̲+)p-1+b(u̲-)p-1+g1(x,u̲)=λ1(εφ1)p-1-a(εφ1)p-1+g1(x,εφ1)≤(λ1-a)(εφ1)p-1+(a-λ1)(εφ1)p-1=0
provided 0≤εφ1≤δa. The latter can be satisfied by choosing ε sufficiently small such that ε∈(0,δa/∥φ1∥∞), where ∥φ1∥∞ stands for the supremum-norm of φ1. This proves that εφ1 is a subsolution if ε∈(0,δa/∥φ1∥∞). In a similar way one can show that for ε sufficiently small the function -εφ1 is a negative supersolution.Applying a recently obtained comparison result that holds for even more general elliptic inclusions (see [38, Theorem 4.1, Corollary 4.1] we immediately obtain the following theorem.Theorem 3.9.
Let hypotheses (g1)-(g2) be satisfied and assume the existence of a subsolutionu̲ and supersolution u¯ of (1.7) such that u̲≤u¯. Then there exist extremal solutions of (1.7) within [u̲,u¯].


### 3.2. Extremal Constant-Sign Solutions and Their Variational Characterization

Combining the results of Lemmas3.6, 3.7, and 3.8 and Theorem 3.9 we immediately deduce the existence of nontrivial positive solutions of problem (1.7) provided the parameter a satisfies a>λ1 that and the existence of negative solutions of problem (1.7) provided that the parameter b satisfies b>λ1. Our main goal of this section is to show that problem (1.7) has a smallest positive solution u+∈int(C01(Ω¯)+) and a greatest negative solution u-∈-int(C01(Ω¯)+). More precisely the following result will be shown.Theorem 3.10.
Let hypotheses (g1)–(g4) be fulfilled. For everya>λ1 and b∈ℝ there exists a smallest positive solution u+=u+(a)∈int(C01(Ω¯)+) of (1.7) within the order interval [0,αae] with the constant αa>0 as in Lemma 3.6. For every b>λ1 and a∈ℝ there exists a greatest negative solution u-=u-(b)∈-int(C01(Ω¯)+) of (1.7) within the order interval [-βbe,0] with the constant βb>0 as in Lemma 3.7.Proof.
Leta>λ1. Lemmas 3.6 and 3.8 ensure that u¯=αae∈int(C01(Ω¯)+) is a supersolution of problem (1.7) and u̲=εφ1∈int(C01(Ω¯)+) is a subsolution of problem (1.7) provided that ε>0 is sufficiently small. We may choose ε>0 such that, in addition, εφ1≤αae. Thus by Theorem 3.9 there exists a smallest and a greatest solution of (1.7) within the ordered interval [εφ1,αae]. Let us denote the smallest solution by uε. Moreover, the nonlinear regularity theory for the p-Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez's strong maximum principle [28] ensure that uε∈int(C01(Ω¯)+). Thus for every positive integer n sufficiently large there is a smallest solution un∈int(C01(Ω¯)+) of problem (1.7) within [(1/n)φ1,αae]. In this way we inductively construct a sequence (un) of smallest solutions which is monotone decreasing; that is, we have
(3.21)un↓u+pointwise
with some function u+:Ω→ℝ satisfying 0≤u+≤αae.Claim 1.
u+ is a solution of problem (1.7).As un∈int(C01(Ω¯)+) and un are solutions of (1.7) we have
(3.22)∥∇un∥pp=∫Ω(aun(x)p-ηn(x))un(x)dx,
where ηn∈Lq′(Ω) and ηn(x)∈∂G(x,un(x)) for almost all x∈Ω. Since un∈[(1/n)φ1,αae], the last equation together with (g2) implies that the sequence (un) is bounded in V0. Taking into account (3.21) we obtain that u+∈V0 and
(3.23)un⇀u+inV0,un→u+inLp(Ω)anda.e.inΩ.
The solution un of (1.7) satisfies
(3.24)〈-Δpun,φ〉=∫Ω(aunp-1-ηn)φdx,∀φ∈V0,
which yields with φ=un-u+ in (3.24) the equation
(3.25)〈-Δpun,un-u+〉=∫Ω(aunp-1-ηn)(un-u+)dx.
Using the convergence properties (3.23) of (un) and (g2) as well as the uniform boundedness of the sequence (un), we get by applying Lebesgue's dominated convergence theorem
(3.26)limn→∞〈-Δpun,un-u+〉=0,
which by the S+-property of -Δp on V0 implies
(3.27)un→u+inV0asn→∞.
Since un are uniformly bounded, from (g2) we see that there exists a constant c>0 such that
(3.28)|ηn(x)|≤ca.e.inΩ,∀n∈ℕ,
and thus we get (for some subsequence if necessary) ηn⇀η+ in Lq′(Ω). By the strong convergence (3.27), Lemma 3.1 can be applied to show that η+(x)∈∂G(x,u+(x)) for almost every x∈Ω. Passing to the limit in (3.24) (for some subsequence if necessary) proves Claim 1.
Asu+ belongs, in particular, to L∞(Ω), Claim 1 and Assumption (g2) implies Δpu+∈L∞(Ω). The nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) ensures that u+∈C1,γ(Ω¯) for some γ∈(0,1), so u+∈C01(Ω¯). In view of (g2) (g3) a constant c̃a>0 can be found such that(3.29)|g(x,s)|≤c̃asp-1fora.a.x∈Ω∀0≤s≤αa∥e∥∞,
which yields in conjunction with Claim 1 that
(3.30)Δpu+≤c̃au+p-1.
We now apply Vázquez's strong maximum principle [28] where in its statement the function β is chosen as β(s)=c̃asp-1 for all s>0, which is possible because ∫0+(1/(sβ(s))1/p)ds=+∞. This result guarantees that if u+≠0, then u+>0 in Ω and ∂u+/∂n<0 on ∂Ω which means that u+∈int(C01(Ω¯)+).Claim 2.
u+∈int(C01(Ω¯)+).Suppose that Claim 2 does not hold. Then by Vázquez's strong maximum principle we must have u+=0, and thus the sequence (un) satisfies
(3.31)un(x)↓0∀x∈Ω.
Setting
(3.32)ũn=un∥∇un∥p∀n,
we may suppose that along a relabelled subsequence one has
(3.33)ũn⇀ũinV0,ũn→ũinLp(Ω)anda.einΩ
with some ũ∈V0, and there is a function w∈Lp(Ω)+ such that
(3.34)|ũn(x)|≤w(x)foralmostallx∈Ω.
Since un are positive solutions of (1.7), we get for ũn the following variational equation:
(3.35)〈-Δpũn,φ〉=a∫Ωũnp-1φdx-∫Ωηnunp-1ũnp-1φdx,∀φ∈V0.
With the special test function φ=ũn-ũ in (3.35) we obtain
(3.36)〈-Δpũn,ũn-ũ〉=a∫Ωũnp-1(ũn-ũ)dx-∫Ωηnunp-1ũnp-1(ũn-ũ)dx.
From (3.29) and (3.34) we get the estimate
(3.37)|ηn(x)|un(x)p-1ũnp-1(x)|ũn(x)-ũ(x)|≤c̃aw(x)p-1(w(x)+|ũ(x)|)fora.a.x∈Ω.
As the right-hand side of the last inequality is in L1(Ω), we may apply Lebesgue's dominated convergence theorem, which in conjunction with (3.33) yields
(3.38)limn→∞∫Ωηnunp-1ũnp-1(ũn-ũ)dx=0.
From (3.33) and (3.36) we conclude
(3.39)limn→∞〈-Δpũn,ũn-ũ〉=0,
which in view of the S+-property of -Δp on V0 results in
(3.40)ũn→ũinV0asn→∞,
and therefore, in particular, ∥∇ũ∥p=1. Taking into account (g3), (3.31), and (3.40), we may pass to the limit in (3.35) which results in
(3.41)〈-Δpũ,φ〉=a∫Ωũp-1φdx,∀φ∈V0.
As ũ≠0, relation (3.41) expresses the fact that ũ≥0 is an eigenfunction of (-Δp,V0) corresponding to the eigenvalue a. As a>λ1, this is impossible according to Anane [30], because ũ must change sign. This contradiction proves that Claim 2 holds true. Note that unlike in the proof of Theorem 2.4, here the contradiction is achieved by the sign-changing property of eigenfunctions belonging to eigenvalues bigger than λ1.Claim 3.
u+∈int(C01(Ω¯)+) is the smallest positive solution of (1.7) in [0,αae].We already know that u+∈[0,αae]. Assume that u∈V0 is any positive solution of (1.7) belonging to [0,αae]. Since u∈L∞(Ω), then by (1.7) and (g3) we deduce Δpu∈L∞(Ω). Using [27, Theorem 1.56] we derive u∈C01(Ω¯), and applying Vázquez's strong maximum principle [28] we infer u∈int(C01(Ω¯)+), which yields u∈[(1/n)φ1,αae] for n sufficiently large. This in conjunction with the fact that un is the least solution of (1.7) in [(1/n)φ1,αae] ensures un≤u if n is large enough. In view of (3.21), we obtain u+≤u, which proves Claim 3.
The proof of the existence of the greatest negative solutionu-=u-(b)∈-int(C01(Ω¯)+) of (1.7) within the ordered interval [-βbe,0] can be done in a similar way. This completes the proof of the theorem.Under hypotheses (g1)–(g4), Theorem3.10 ensures the existence of extremal positive and negative solutions of (1.7) for all a>λ1 and b>λ1 denoted by u+=u+(a)∈int(C01(Ω¯)+) and u-=u-(b)∈-int(C01(Ω¯)+), respectively. In what follows we are going to characterize these extremal solutions as global (local) minimizers of certain locally Lipschitz functionals that are generated by truncation procedures. To this end let us introduce truncation functions related to the extremal solutions u+ and u- as follows: (3.42)τ+(x,s)={0ifs<0,sif0≤s≤u+(x),u+(x)ifs>u+(x),τ-(x,s)={u-(x)ifs<u-(x),sifu-(x)≤s≤0,0ifs>0,τ0(x,s)={u-(x)ifs<u-(x),sifu-(x)≤s≤u+(x),u+(x)ifs>u+(x).
The truncations τ+,τ-,τ0:Ω×ℝ→ℝ are continuous, uniformly bounded, and Lipschitzian with respect to s. The extremal positive and negative solutions u+ and u- of (1.7), respectively, ensured by Theorem 3.10 satisfy(3.43)-Δpu+=a(u+)p-1-η+,-Δpu-=-b(u-)p-1-η-inV0*,
where η+,η-∈Lq′(Ω) and (3.44)η+(x)∈∂G(x,u+(x)),η-(x)∈∂G(x,u-(x))
for a.a. x∈Ω. By means of η+,η- we introduce the following truncations of the nonlinearity g:Ω×ℝ→ℝ: (3.45)g+(x,s):={0ifs<0,g(x,s)for0≤s≤u+(x),η+(x)whens>u+(x),g-(x,s):={η-(x)ifs<u-(x),g(x,s)foru-(x)≤s≤0,0whens>0,g0(x,s):={η-(x)ifs<u-(x),g(x,s)foru-(x)≤s≤u+(x),η+(x)whens>u+(x)
and define functionals E+,E-,E0 by (3.46)E+(u)=1p∥∇u∥pp-∫Ω∫0u(x)(aτ+(x,s)p-1-g+(x,s))dsdx,E-(u)=1p∥∇u∥pp+∫Ω∫0u(x)(b|τ-(x,s)|p-1+g-(x,s))dsdx,E0(u)=1p∥∇u∥pp-∫Ω∫0u(x)(aτ+(x,s)p-1-b|τ-(x,s)|p-1-g0(x,s))dsdx.
Due to (g2) the functionals E+,E-,E0:V0→ℝ are locally Lipschitz continuous. Moreover, in view of the truncations involved these functionals are bounded below, coercive, and weakly lower semicontinuous such that their global minimizers exist. The following lemma provides a characterization of the critical points of these functionals.Lemma 3.11.
Letu+ and u- be the extremal constant-sign solutions of (1.7). Then the following holds. (i)
A critical pointv∈V0 of E+ is a (nonnegative) solution of (1.7) satisfying 0≤v≤u+.(ii)
A critical pointw∈V0 of E- is a (nonpositive) solution of (1.7) satisfying u-≤w≤0.(iii)
A critical pointz∈V0 of E0 is a solution of (1.7) satisfying u-≤z≤u+.Proof.
To prove (i) letv be a critical point of E+, that is, 0∈∂E+(v), which results in
(3.47)v∈V0:-Δpv=aτ+(x,v)p-1-winV0*
for some w∈Lq′(Ω) and such that w(x)∈∂G+(x,v(x)) almost everywhere in Ω, with
(3.48)G+(x,ξ):=∫0ξg+(x,t)dt,(x,ξ)∈Ω×ℝ.
Let us show that v≤u+ holds. As u+ is a positive solution of (1.7), it satisfies the first equation in (3.43), and by subtracting that equation from (3.47) and applying the special test function φ=(v-u+)+ we get
(3.49)∫Ω(|∇v|p-2∇v-|∇u+|p-2∇u+)∇(v-u+)+dx=∫Ω[a(τ+(x,v)p-1-u+p-1)-(w-η+)](v-u+)+dx.
By the definition of the truncations introduced above we have τ+(x,v)p-1(x)-u+p-1(x)=0 and w(x)-η+(x)=0 for a.a. x∈{v>u+}, and thus the right-hand side of (3.49) is zero which leads to
(3.50)∫Ω(|∇v|p-2∇v-|∇u+|p-2∇u+)∇(v-u+)+dx=0,
and hence it follows ∇(v-u+)+=0. Because (v-u+)+∈V0, this implies (v-u+)+=0 which proves v≤u+. To prove 0≤v we test (3.47) with φ=v-=max{-v,0}∈V0 and get
(3.51)∫Ω|∇v|p-2∇v∇v-dx=∫Ω(aτ+(x,v)p-1-w)v-dx=0,
which results in ∥∇v-∥pp=0, and thus v-=0, that is, 0≤v. Thus the critical point v of E+ which is a solution of the Dirichlet problem (3.47) satisfies 0≤v≤u+, and therefore τ+(x,v)=v. Because ∂G+(x,v(x))⊂∂G(x,v(x)), it follows w∈∂G(x,v(x)), and therefore v must be a solution of (1.7). This proves (i). In just the same way one can prove also (ii) and (iii) which is omitted.The following lemma provides a variational characterization of the extremal constant-sign solutionsu+ and u-.Lemma 3.12.
Leta>λ1 and b>λ1. Then the extremal positive solution u+ of (1.7) is the unique global minimizer of the functional E+, and the extremal negative solution u- of (1.7) is the unique global minimizer of the functional E-. Both u+ and u- are local minimizers of E0.Proof.
The functionalE+:V0→ℝ is bounded below, coercive, and weakly lower semicontinuous. Thus there exists a global minimizer v+∈V0 of E+, that is,
(3.52)E+(v+)=infu∈V0E+(u),
As v+ is a critical point of E+, so by Lemma 3.11 it is a nonnegative solution of (1.7) satisfying 0≤v+≤u+. Since a-λ1>0, there is a νa>0 such that a-λ1-νa>0. By (g3) we infer the existence of a δ̃a>0 such that
(3.53)|g(x,s)|≤(a-λ1-νa)sp-1,∀s:0<s≤δ̃a,
and thus for ε>0 sufficiently small such that
(3.54)εφ1≤u+,ε∥φ1∥L∞(Ω)≤δ̃a,
we obtain (note ∥φ1∥p=1)
(3.55)E+(v+)≤E+(εφ1)=λ1pεp+∫Ω∫0εφ1(x)(-asp-1+g(x,s))dsdx≤λ1pεp+∫Ω∫0εφ1(x)(-λ1-νa)sp-1dsdx<0.
Hence it follows that E+(v+)<0, and thus v+≠0. Applying nonlinear regularity theory for the p-Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez's strong maximum principle, we see that v+∈int(C01(Ω¯)+). As u+ is the smallest positive solution of (1.7) in [0,αae] and 0≤v+≤u+, it follows v+=u+, which shows that the global minimizer v+ must be unique and equal to u+. By similar arguments one can show that the global minimizer v- of E- must be unique and v-=u-. It remains to prove that u+ and u- are local minimizers of E0. Let us show this last assertion for u+ only. By definition we have
(3.56)E+(u)=E0(u)∀u∈V0withu≥0.
Since u+ is a global minimizer of E+ and u+∈int(C01(Ω¯)+), it follows that u+ is a local minimizer of E0 with respect to the C1 topology. Due to a result by Motreanu and Papageorgiou in [39, Proposition 4], we conclude that u+ is also a local minimizer of E0 with respect to the V0 topology. This completes the proof of the lemma.Lemma 3.13.
The functionalE0:V0→ℝ has a global minimizer v0 which is a nontrivial solution of (1.7) satisfying u-≤v0≤u+.Proof.
One easily verifies thatE0:V0→ℝ is coercive and weakly lower semicontinuous, and thus a global minimizer v0 exists which is a critical point of E0. Apply Lemma 3.11(iii) and note that, for example, E0(u+)=E+(u+)<0, which shows that v0≠0.


### 3.3. Sign-Changing Solutions

Theorem3.10 ensures the existence of a smallest positive solution u+∈int(C01(Ω¯)+) in [0,αae] and a greatest negative solution u-∈-int(C01(Ω¯)+) of (1.7) in [-βbe,0]. The idea to show the existence of sign-changing solutions is to prove the existence of nontrivial solutions u0 of (1.7) satisfying u-≤u0≤u+ with u0≠u- and u0≠u+, which then must be sign-changing, because u+ and u- are the extremal constant-sign solutions.Theorem 3.14.
Let hypotheses (g1)–(g4) and (H) be satisfied. Then problem (1.7) has a smallest positive solution u+∈int(C01(Ω¯)+) in [0,αae], a greatest negative solution u-∈-int(C01(Ω¯)+) in [-βbe,0], and a nontrivial sign-changing solution u0∈C01(Ω¯) with u-≤u0≤u+.Proof.
Clearly the existence of the extremal positive and negative solutionu+ and u- follows from Theorem 3.10, because (H), in particular, implies that a>λ1 and b>λ1. As for the existence of a sign-changing solution we first note that by Lemma 3.13 it follows that the global minimizer v0 of E0 is a nontrivial solution of (1.7) satisfying u-≤v0≤u+. Therefore, if v0≠u+ and v0≠u-, then v0=u0 must be a sign-changing solution as asserted, because u- is the greatest negative and u+ is the smallest positive solution of (1.7). Thus, we still need to prove the existence of sign-changing solutions in case that either v0=u- or v0=u+.
Let us consider the casev0=u+ only, since the case v0=u- can be treated quite similarly. By Lemma 3.12, u- is a local minimizer of E0. Without loss of generality we may even assume that u- is a strict local minimizer of E0, because on the contrary we would find (infinitely many) critical points z of E0 that are sign-changing solutions thanks to u-≤z≤u+ and the extremality of the solutions u-, u+ obtained in Theorem 3.10 which proves the assertion.
Therefore, it remains to prove the existence of sign-changing solutions under the assumptions that the global minimizerv0 of E0 is equal to u+, and u- is a strict local minimizer of E0. This implies the existence of ρ∈(0,∥u--u+∥) such that(3.57)E0(u+)≤E0(u-)<inf{E0(u):u∈∂Bρ(u-)},
where ∂Bρ(u-)={u∈V0:∥u-u-∥=ρ}. The functional E0 satisfies the Palais-Smale condition, because it is bounded below, locally Lipschitz continuous, and coercive; see, for example, [40, Corollary 2.4]. Thus in view of (3.57) we may apply the nonsmooth version of Ambrosetti-Rabinowitz's Mountain-Pass Theorem (see, e.g., [41, Theorem 3.4]) which ensures the existence of a critical point u0∈V0 satisfying 0∈∂E0(u0) and
(3.58)inf{E0(u):u∈∂Bρ(u-)}≤E0(u0)=infγ∈Γmaxt∈[-1,1]E0(γ(t)),
where
(3.59)Γ={γ∈C([-1,1],V0):γ(-1)=u-,γ(1)=u+}.
It is clear from (3.57) and (3.58) that u0≠u-and u0≠u+, and thus u0 is a sign-changing solution provided u0≠0. To prove the latter we claim
(3.60)E0(u0)<0
for which it suffices to construct a path γ̂∈Γ such that
(3.61)E0(γ̂(t))<0∀t∈[-1,1].
The construction of such a path γ̂ can be done by adopting an approach due to the authors in [3] and applying the Second Deformation Lemma for locally Lipschitz functionals as it can be found in [42, Theorem 2.10]. This completes the proof.Remark 3.15.
The multiplicity results and the existence of sign-changing solutions obtained in this section generalize very recent results due to the authors obtained in [3, 4, 7]. Moreover, if the function t↦g(x,t) is continuous on ℝ and a=b=λ, then ∂G(x,ξ)={g(x,ξ)}, and problem (1.7) reduces to
(3.62)u∈V0:-Δpu=λ|u|p-2u-g(x,u)inV0*.
However, even in this setting the results obtained here are more general than obtained in [6, Theorem 3.9], because we do not assume that g(x,t)t≥0 for all t∈ℝ.Remark 3.16.
Theorem3.14 improves also Corollary 3.2 of [8]. In fact, let p=2, let u∈V0 be a solution of (1.7) in case a=b=λ and g(x,t)≡g(t), (x,t)∈Ω×ℝ with η∈Lq′(Ω) satisfying η(x)∈∂G(u(x)). By definition of Clarke's gradient we have, for any φ∈V0,
(3.63)η(x)φ(x)≤G0(u(x);φ(x))a.e.inΩ.
As u is a solution, the following holds: u∈V0 and (p=2),
(3.64)∫Ω∇u∇φdx=λ∫Ωuφdx-∫Ωηφdx,
which yields
(3.65)-∫Ω∇u∇φdx+λ∫Ωuφdx=∫Ωηφdx≤∫ΩG0(u;φ)dx,∀φ∈V0,
That is, u turns out to be a solution of the hemivariational inequality studied in [8]. Since the hypotheses of [8, Corollary 3.2] imply (g1)–(g4), the assertion follows.Remark 3.17.
Multiplicity results for a nonsmooth version of problem (1.4) in form of (1.2) can be established under structure conditions for Clarke's gradient ∂j similar to H(f).
Multiplicity and sign-changing solutions results have been obtained recently in [43] for the following nonlinear Neumann-type boundary value problem: find u∈V∖{0} and parameters a,b∈ℝ such that(3.66)-Δpu=f(x,u)-|u|p-2uinΩ,|∇u|p-2∂u∂ν=a(u+)p-1-b(u-)p-1+g(x,u)on∂Ω.
For problem (3.66) conditions on the parameters have been given in terms of the “Steklov-Fučik” spectrum to ensure multiplicity results.


## 3.1. Hypotheses, Definitions, and Preliminaries

We impose the following hypotheses on the nonlinearityg:Ω×ℝ→ℝ whose primitive is G of problem (1.7)(g1)
(x,s)↦g(x,s) is measurable in each variable separately.(g2)
There existsc>0, and q∈[p,p*) such that
(3.3)|g(x,s)|≤c(1+|s|q-1)
for a.a. x∈Ω and for all s∈ℝ, where p* denotes the critical Sobolev exponent which is p*=Np/(N-p) if p<N, and p*=+∞ if p≥N.(g3)
One has(3.4)lims→0g(x,s)|s|p-2s=0uniformlywithrespecttoa.a.x∈Ω.(g4)
One has(3.5)lim|s|→∞g(x,s)|s|p-2s=+∞uniformlywithrespecttoa.a.x∈Ω.In view of assumptions (g1) and (g2) the functions↦G(x,s) is locally Lipschitz and the functional 𝒢:Lq(Ω)→ℝ defined by(3.6)𝒢(u):=∫ΩG(x,u(x))dx,u∈Lq(Ω)
is well defined and locally Lipschitz continuous as well. The generalized gradients ∂G(x,·) and ∂𝒢 can be characterized as follows: Define for every (x,t)∈Ω×ℝ, (3.7)g1(x,t):=limδ→0+essinf|τ-t|<δg(x,τ),g2(x,t):=limδ→0+esssup|τ-t|<δg(x,τ).
Proposition 1.7 in [35] ensures that (3.8)∂G(x,ξ)=[g1(x,ξ),g2(x,ξ)],
while Theorem 4.5.19 of [36] implies (3.9)∂𝒢(u)⊆{w∈Lq′(Ω):g1(x,u(x))≤w(x)≤g2(x,u(x))a.e.inΩ}
with q′:=q/(q-1). The next result is an immediate consequence of [37, Proposition 2.1.5].Lemma 3.1.
Supposeun→u in V0, wn⇀w in Lq′(Ω), and wn∈∂𝒢(un) for all n∈ℕ. Then w∈∂𝒢(u).Definition 3.2.
A functionu∈V0 is called a solution of (1.7) if there is an η∈Lq′(Ω) such that
(3.10)η(x)∈∂G(x,u(x))fora.a.x∈Ω,∫Ω|∇u|p-2∇u∇φdx+∫Ω(η-a(u+)p-1+b(u-)p-1)φdx=0,∀φ∈V0.Remark 3.3.
Due to assumption (g3) we haveg1(x,0)≤0≤g2(x,0) for almost all x∈Ω. Hence, in view of (3.8), problem (1.7) always possesses the trivial solution.Definition 3.4.
A functionu̲∈V:=W1,p(Ω) is called a subsolution of (1.7) if u̲|∂Ω≤0, and if there is an η̲∈Lq′(Ω) such that
(3.11)η̲(x)∈∂G(x,u̲(x))fora.a.x∈Ω,∫Ω|∇u̲|p-2∇u̲∇φdx+∫Ω(η̲-a(u̲+)p-1+b(u̲-)p-1)φdx≤0,∀φ∈V0∩Lp(Ω)+.Similarly, we define a supersolution as follows.Definition 3.5.
A functionu¯∈V is called a supersolution of (1.7) if u¯|∂Ω≥0, and if there is an η¯∈Lq′(Ω) such that
(3.12)η¯(x)∈∂G(x,u¯(x))fora.a.x∈Ω,∫Ω|∇u¯|p-2∇u¯∇φdx+∫Ω(η¯-a(u¯+)p-1+b(u¯-)p-1)φdx≥0,∀φ∈V0∩Lp(Ω)+.Lemma 3.6.
Lete be the uniquely defined solution of (2.6). If a>λ1, then there exists a constant αa>0 such that for any b∈ℝ the function αae is a positive supersolution of problem (1.7).Proof.
Leta>λ1. By (g4) there is sa>0 such that
(3.13)g(x,s)sp-1>afora.a.x∈Ω∀s>sa,
and by (g2) we get
(3.14)|g(x,s)-asp-1|≤|g(x,s)|+asp-1≤ca,fora.a.x∈Ω∀s∈[0,sa],
which implies
(3.15)g(x,s)≥asp-1-cafora.a.x∈Ω∀s≥0,
and thus in view of the definition of g2 we obtain
(3.16)g2(x,s)≥asp-1-cafora.a.x∈Ω∀s≥0,
Let u̅=αae, where αa is a positive constant to be specified. Then we get
(3.17)-Δpu̅-a(u̅+)p-1+b(u̅-)p-1+g2(x,u̅)=αap-1-aαap-1ep-1+g2(x,αae)≥αap-1-ca,
which shows that for αa:=ca1/(p-1) the function αae is in fact a supersolution of (1.7) with η¯(x)=g2(x,u̅(x)).In a similar way the following lemma on the existence of a negative subsolution can be proved.Lemma 3.7.
Lete be the uniquely defined solution of (2.6). If b>λ1, then there exists a constant βb>0 such that for any a∈ℝ the function -βbe is a negative subsolution of problem (1.7).In the next lemma we demonstrate that small constant multiples ofφ1 may be sub- and supersolutions of (1.7). More precisely we have the following result.Lemma 3.8.
Letφ1 be the normalized positive eigenfunction corresponding to the first eigenvalue λ1 of (-Δp,V0). If a>λ1, then for ε>0 sufficiently small and any b∈ℝ the function εφ1 is a positive subsolution of problem (1.7). If b>λ1, then for ε>0 sufficiently small and any a∈ℝ the function -εφ1 is a negative supersolution of problem (1.7).Proof.
By (g3) there is a constantδa>0 such that
(3.18)|g(x,s)||s|p-1<a-λ1fora.a.x∈Ω∀0<|s|≤δa,
which implies
(3.19)|g(x,s)|≤(a-λ1)sp-1fora.a.x∈Ω,∀s:0≤s≤δa.
Define u̲=εφ1 with ε>0. Applying (3.19) and the definition of g1 we get
(3.20)-Δpu̲-a(u̲+)p-1+b(u̲-)p-1+g1(x,u̲)=λ1(εφ1)p-1-a(εφ1)p-1+g1(x,εφ1)≤(λ1-a)(εφ1)p-1+(a-λ1)(εφ1)p-1=0
provided 0≤εφ1≤δa. The latter can be satisfied by choosing ε sufficiently small such that ε∈(0,δa/∥φ1∥∞), where ∥φ1∥∞ stands for the supremum-norm of φ1. This proves that εφ1 is a subsolution if ε∈(0,δa/∥φ1∥∞). In a similar way one can show that for ε sufficiently small the function -εφ1 is a negative supersolution.Applying a recently obtained comparison result that holds for even more general elliptic inclusions (see [38, Theorem 4.1, Corollary 4.1] we immediately obtain the following theorem.Theorem 3.9.
Let hypotheses (g1)-(g2) be satisfied and assume the existence of a subsolutionu̲ and supersolution u¯ of (1.7) such that u̲≤u¯. Then there exist extremal solutions of (1.7) within [u̲,u¯].


## 3.2. Extremal Constant-Sign Solutions and Their Variational Characterization

Combining the results of Lemmas3.6, 3.7, and 3.8 and Theorem 3.9 we immediately deduce the existence of nontrivial positive solutions of problem (1.7) provided the parameter a satisfies a>λ1 that and the existence of negative solutions of problem (1.7) provided that the parameter b satisfies b>λ1. Our main goal of this section is to show that problem (1.7) has a smallest positive solution u+∈int(C01(Ω¯)+) and a greatest negative solution u-∈-int(C01(Ω¯)+). More precisely the following result will be shown.Theorem 3.10.
Let hypotheses (g1)–(g4) be fulfilled. For everya>λ1 and b∈ℝ there exists a smallest positive solution u+=u+(a)∈int(C01(Ω¯)+) of (1.7) within the order interval [0,αae] with the constant αa>0 as in Lemma 3.6. For every b>λ1 and a∈ℝ there exists a greatest negative solution u-=u-(b)∈-int(C01(Ω¯)+) of (1.7) within the order interval [-βbe,0] with the constant βb>0 as in Lemma 3.7.Proof.
Leta>λ1. Lemmas 3.6 and 3.8 ensure that u¯=αae∈int(C01(Ω¯)+) is a supersolution of problem (1.7) and u̲=εφ1∈int(C01(Ω¯)+) is a subsolution of problem (1.7) provided that ε>0 is sufficiently small. We may choose ε>0 such that, in addition, εφ1≤αae. Thus by Theorem 3.9 there exists a smallest and a greatest solution of (1.7) within the ordered interval [εφ1,αae]. Let us denote the smallest solution by uε. Moreover, the nonlinear regularity theory for the p-Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez's strong maximum principle [28] ensure that uε∈int(C01(Ω¯)+). Thus for every positive integer n sufficiently large there is a smallest solution un∈int(C01(Ω¯)+) of problem (1.7) within [(1/n)φ1,αae]. In this way we inductively construct a sequence (un) of smallest solutions which is monotone decreasing; that is, we have
(3.21)un↓u+pointwise
with some function u+:Ω→ℝ satisfying 0≤u+≤αae.Claim 1.
u+ is a solution of problem (1.7).As un∈int(C01(Ω¯)+) and un are solutions of (1.7) we have
(3.22)∥∇un∥pp=∫Ω(aun(x)p-ηn(x))un(x)dx,
where ηn∈Lq′(Ω) and ηn(x)∈∂G(x,un(x)) for almost all x∈Ω. Since un∈[(1/n)φ1,αae], the last equation together with (g2) implies that the sequence (un) is bounded in V0. Taking into account (3.21) we obtain that u+∈V0 and
(3.23)un⇀u+inV0,un→u+inLp(Ω)anda.e.inΩ.
The solution un of (1.7) satisfies
(3.24)〈-Δpun,φ〉=∫Ω(aunp-1-ηn)φdx,∀φ∈V0,
which yields with φ=un-u+ in (3.24) the equation
(3.25)〈-Δpun,un-u+〉=∫Ω(aunp-1-ηn)(un-u+)dx.
Using the convergence properties (3.23) of (un) and (g2) as well as the uniform boundedness of the sequence (un), we get by applying Lebesgue's dominated convergence theorem
(3.26)limn→∞〈-Δpun,un-u+〉=0,
which by the S+-property of -Δp on V0 implies
(3.27)un→u+inV0asn→∞.
Since un are uniformly bounded, from (g2) we see that there exists a constant c>0 such that
(3.28)|ηn(x)|≤ca.e.inΩ,∀n∈ℕ,
and thus we get (for some subsequence if necessary) ηn⇀η+ in Lq′(Ω). By the strong convergence (3.27), Lemma 3.1 can be applied to show that η+(x)∈∂G(x,u+(x)) for almost every x∈Ω. Passing to the limit in (3.24) (for some subsequence if necessary) proves Claim 1.
Asu+ belongs, in particular, to L∞(Ω), Claim 1 and Assumption (g2) implies Δpu+∈L∞(Ω). The nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) ensures that u+∈C1,γ(Ω¯) for some γ∈(0,1), so u+∈C01(Ω¯). In view of (g2) (g3) a constant c̃a>0 can be found such that(3.29)|g(x,s)|≤c̃asp-1fora.a.x∈Ω∀0≤s≤αa∥e∥∞,
which yields in conjunction with Claim 1 that
(3.30)Δpu+≤c̃au+p-1.
We now apply Vázquez's strong maximum principle [28] where in its statement the function β is chosen as β(s)=c̃asp-1 for all s>0, which is possible because ∫0+(1/(sβ(s))1/p)ds=+∞. This result guarantees that if u+≠0, then u+>0 in Ω and ∂u+/∂n<0 on ∂Ω which means that u+∈int(C01(Ω¯)+).Claim 2.
u+∈int(C01(Ω¯)+).Suppose that Claim 2 does not hold. Then by Vázquez's strong maximum principle we must have u+=0, and thus the sequence (un) satisfies
(3.31)un(x)↓0∀x∈Ω.
Setting
(3.32)ũn=un∥∇un∥p∀n,
we may suppose that along a relabelled subsequence one has
(3.33)ũn⇀ũinV0,ũn→ũinLp(Ω)anda.einΩ
with some ũ∈V0, and there is a function w∈Lp(Ω)+ such that
(3.34)|ũn(x)|≤w(x)foralmostallx∈Ω.
Since un are positive solutions of (1.7), we get for ũn the following variational equation:
(3.35)〈-Δpũn,φ〉=a∫Ωũnp-1φdx-∫Ωηnunp-1ũnp-1φdx,∀φ∈V0.
With the special test function φ=ũn-ũ in (3.35) we obtain
(3.36)〈-Δpũn,ũn-ũ〉=a∫Ωũnp-1(ũn-ũ)dx-∫Ωηnunp-1ũnp-1(ũn-ũ)dx.
From (3.29) and (3.34) we get the estimate
(3.37)|ηn(x)|un(x)p-1ũnp-1(x)|ũn(x)-ũ(x)|≤c̃aw(x)p-1(w(x)+|ũ(x)|)fora.a.x∈Ω.
As the right-hand side of the last inequality is in L1(Ω), we may apply Lebesgue's dominated convergence theorem, which in conjunction with (3.33) yields
(3.38)limn→∞∫Ωηnunp-1ũnp-1(ũn-ũ)dx=0.
From (3.33) and (3.36) we conclude
(3.39)limn→∞〈-Δpũn,ũn-ũ〉=0,
which in view of the S+-property of -Δp on V0 results in
(3.40)ũn→ũinV0asn→∞,
and therefore, in particular, ∥∇ũ∥p=1. Taking into account (g3), (3.31), and (3.40), we may pass to the limit in (3.35) which results in
(3.41)〈-Δpũ,φ〉=a∫Ωũp-1φdx,∀φ∈V0.
As ũ≠0, relation (3.41) expresses the fact that ũ≥0 is an eigenfunction of (-Δp,V0) corresponding to the eigenvalue a. As a>λ1, this is impossible according to Anane [30], because ũ must change sign. This contradiction proves that Claim 2 holds true. Note that unlike in the proof of Theorem 2.4, here the contradiction is achieved by the sign-changing property of eigenfunctions belonging to eigenvalues bigger than λ1.Claim 3.
u+∈int(C01(Ω¯)+) is the smallest positive solution of (1.7) in [0,αae].We already know that u+∈[0,αae]. Assume that u∈V0 is any positive solution of (1.7) belonging to [0,αae]. Since u∈L∞(Ω), then by (1.7) and (g3) we deduce Δpu∈L∞(Ω). Using [27, Theorem 1.56] we derive u∈C01(Ω¯), and applying Vázquez's strong maximum principle [28] we infer u∈int(C01(Ω¯)+), which yields u∈[(1/n)φ1,αae] for n sufficiently large. This in conjunction with the fact that un is the least solution of (1.7) in [(1/n)φ1,αae] ensures un≤u if n is large enough. In view of (3.21), we obtain u+≤u, which proves Claim 3.
The proof of the existence of the greatest negative solutionu-=u-(b)∈-int(C01(Ω¯)+) of (1.7) within the ordered interval [-βbe,0] can be done in a similar way. This completes the proof of the theorem.Under hypotheses (g1)–(g4), Theorem3.10 ensures the existence of extremal positive and negative solutions of (1.7) for all a>λ1 and b>λ1 denoted by u+=u+(a)∈int(C01(Ω¯)+) and u-=u-(b)∈-int(C01(Ω¯)+), respectively. In what follows we are going to characterize these extremal solutions as global (local) minimizers of certain locally Lipschitz functionals that are generated by truncation procedures. To this end let us introduce truncation functions related to the extremal solutions u+ and u- as follows: (3.42)τ+(x,s)={0ifs<0,sif0≤s≤u+(x),u+(x)ifs>u+(x),τ-(x,s)={u-(x)ifs<u-(x),sifu-(x)≤s≤0,0ifs>0,τ0(x,s)={u-(x)ifs<u-(x),sifu-(x)≤s≤u+(x),u+(x)ifs>u+(x).
The truncations τ+,τ-,τ0:Ω×ℝ→ℝ are continuous, uniformly bounded, and Lipschitzian with respect to s. The extremal positive and negative solutions u+ and u- of (1.7), respectively, ensured by Theorem 3.10 satisfy(3.43)-Δpu+=a(u+)p-1-η+,-Δpu-=-b(u-)p-1-η-inV0*,
where η+,η-∈Lq′(Ω) and (3.44)η+(x)∈∂G(x,u+(x)),η-(x)∈∂G(x,u-(x))
for a.a. x∈Ω. By means of η+,η- we introduce the following truncations of the nonlinearity g:Ω×ℝ→ℝ: (3.45)g+(x,s):={0ifs<0,g(x,s)for0≤s≤u+(x),η+(x)whens>u+(x),g-(x,s):={η-(x)ifs<u-(x),g(x,s)foru-(x)≤s≤0,0whens>0,g0(x,s):={η-(x)ifs<u-(x),g(x,s)foru-(x)≤s≤u+(x),η+(x)whens>u+(x)
and define functionals E+,E-,E0 by (3.46)E+(u)=1p∥∇u∥pp-∫Ω∫0u(x)(aτ+(x,s)p-1-g+(x,s))dsdx,E-(u)=1p∥∇u∥pp+∫Ω∫0u(x)(b|τ-(x,s)|p-1+g-(x,s))dsdx,E0(u)=1p∥∇u∥pp-∫Ω∫0u(x)(aτ+(x,s)p-1-b|τ-(x,s)|p-1-g0(x,s))dsdx.
Due to (g2) the functionals E+,E-,E0:V0→ℝ are locally Lipschitz continuous. Moreover, in view of the truncations involved these functionals are bounded below, coercive, and weakly lower semicontinuous such that their global minimizers exist. The following lemma provides a characterization of the critical points of these functionals.Lemma 3.11.
Letu+ and u- be the extremal constant-sign solutions of (1.7). Then the following holds. (i)
A critical pointv∈V0 of E+ is a (nonnegative) solution of (1.7) satisfying 0≤v≤u+.(ii)
A critical pointw∈V0 of E- is a (nonpositive) solution of (1.7) satisfying u-≤w≤0.(iii)
A critical pointz∈V0 of E0 is a solution of (1.7) satisfying u-≤z≤u+.Proof.
To prove (i) letv be a critical point of E+, that is, 0∈∂E+(v), which results in
(3.47)v∈V0:-Δpv=aτ+(x,v)p-1-winV0*
for some w∈Lq′(Ω) and such that w(x)∈∂G+(x,v(x)) almost everywhere in Ω, with
(3.48)G+(x,ξ):=∫0ξg+(x,t)dt,(x,ξ)∈Ω×ℝ.
Let us show that v≤u+ holds. As u+ is a positive solution of (1.7), it satisfies the first equation in (3.43), and by subtracting that equation from (3.47) and applying the special test function φ=(v-u+)+ we get
(3.49)∫Ω(|∇v|p-2∇v-|∇u+|p-2∇u+)∇(v-u+)+dx=∫Ω[a(τ+(x,v)p-1-u+p-1)-(w-η+)](v-u+)+dx.
By the definition of the truncations introduced above we have τ+(x,v)p-1(x)-u+p-1(x)=0 and w(x)-η+(x)=0 for a.a. x∈{v>u+}, and thus the right-hand side of (3.49) is zero which leads to
(3.50)∫Ω(|∇v|p-2∇v-|∇u+|p-2∇u+)∇(v-u+)+dx=0,
and hence it follows ∇(v-u+)+=0. Because (v-u+)+∈V0, this implies (v-u+)+=0 which proves v≤u+. To prove 0≤v we test (3.47) with φ=v-=max{-v,0}∈V0 and get
(3.51)∫Ω|∇v|p-2∇v∇v-dx=∫Ω(aτ+(x,v)p-1-w)v-dx=0,
which results in ∥∇v-∥pp=0, and thus v-=0, that is, 0≤v. Thus the critical point v of E+ which is a solution of the Dirichlet problem (3.47) satisfies 0≤v≤u+, and therefore τ+(x,v)=v. Because ∂G+(x,v(x))⊂∂G(x,v(x)), it follows w∈∂G(x,v(x)), and therefore v must be a solution of (1.7). This proves (i). In just the same way one can prove also (ii) and (iii) which is omitted.The following lemma provides a variational characterization of the extremal constant-sign solutionsu+ and u-.Lemma 3.12.
Leta>λ1 and b>λ1. Then the extremal positive solution u+ of (1.7) is the unique global minimizer of the functional E+, and the extremal negative solution u- of (1.7) is the unique global minimizer of the functional E-. Both u+ and u- are local minimizers of E0.Proof.
The functionalE+:V0→ℝ is bounded below, coercive, and weakly lower semicontinuous. Thus there exists a global minimizer v+∈V0 of E+, that is,
(3.52)E+(v+)=infu∈V0E+(u),
As v+ is a critical point of E+, so by Lemma 3.11 it is a nonnegative solution of (1.7) satisfying 0≤v+≤u+. Since a-λ1>0, there is a νa>0 such that a-λ1-νa>0. By (g3) we infer the existence of a δ̃a>0 such that
(3.53)|g(x,s)|≤(a-λ1-νa)sp-1,∀s:0<s≤δ̃a,
and thus for ε>0 sufficiently small such that
(3.54)εφ1≤u+,ε∥φ1∥L∞(Ω)≤δ̃a,
we obtain (note ∥φ1∥p=1)
(3.55)E+(v+)≤E+(εφ1)=λ1pεp+∫Ω∫0εφ1(x)(-asp-1+g(x,s))dsdx≤λ1pεp+∫Ω∫0εφ1(x)(-λ1-νa)sp-1dsdx<0.
Hence it follows that E+(v+)<0, and thus v+≠0. Applying nonlinear regularity theory for the p-Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez's strong maximum principle, we see that v+∈int(C01(Ω¯)+). As u+ is the smallest positive solution of (1.7) in [0,αae] and 0≤v+≤u+, it follows v+=u+, which shows that the global minimizer v+ must be unique and equal to u+. By similar arguments one can show that the global minimizer v- of E- must be unique and v-=u-. It remains to prove that u+ and u- are local minimizers of E0. Let us show this last assertion for u+ only. By definition we have
(3.56)E+(u)=E0(u)∀u∈V0withu≥0.
Since u+ is a global minimizer of E+ and u+∈int(C01(Ω¯)+), it follows that u+ is a local minimizer of E0 with respect to the C1 topology. Due to a result by Motreanu and Papageorgiou in [39, Proposition 4], we conclude that u+ is also a local minimizer of E0 with respect to the V0 topology. This completes the proof of the lemma.Lemma 3.13.
The functionalE0:V0→ℝ has a global minimizer v0 which is a nontrivial solution of (1.7) satisfying u-≤v0≤u+.Proof.
One easily verifies thatE0:V0→ℝ is coercive and weakly lower semicontinuous, and thus a global minimizer v0 exists which is a critical point of E0. Apply Lemma 3.11(iii) and note that, for example, E0(u+)=E+(u+)<0, which shows that v0≠0.


## 3.3. Sign-Changing Solutions

Theorem3.10 ensures the existence of a smallest positive solution u+∈int(C01(Ω¯)+) in [0,αae] and a greatest negative solution u-∈-int(C01(Ω¯)+) of (1.7) in [-βbe,0]. The idea to show the existence of sign-changing solutions is to prove the existence of nontrivial solutions u0 of (1.7) satisfying u-≤u0≤u+ with u0≠u- and u0≠u+, which then must be sign-changing, because u+ and u- are the extremal constant-sign solutions.Theorem 3.14.
Let hypotheses (g1)–(g4) and (H) be satisfied. Then problem (1.7) has a smallest positive solution u+∈int(C01(Ω¯)+) in [0,αae], a greatest negative solution u-∈-int(C01(Ω¯)+) in [-βbe,0], and a nontrivial sign-changing solution u0∈C01(Ω¯) with u-≤u0≤u+.Proof.
Clearly the existence of the extremal positive and negative solutionu+ and u- follows from Theorem 3.10, because (H), in particular, implies that a>λ1 and b>λ1. As for the existence of a sign-changing solution we first note that by Lemma 3.13 it follows that the global minimizer v0 of E0 is a nontrivial solution of (1.7) satisfying u-≤v0≤u+. Therefore, if v0≠u+ and v0≠u-, then v0=u0 must be a sign-changing solution as asserted, because u- is the greatest negative and u+ is the smallest positive solution of (1.7). Thus, we still need to prove the existence of sign-changing solutions in case that either v0=u- or v0=u+.
Let us consider the casev0=u+ only, since the case v0=u- can be treated quite similarly. By Lemma 3.12, u- is a local minimizer of E0. Without loss of generality we may even assume that u- is a strict local minimizer of E0, because on the contrary we would find (infinitely many) critical points z of E0 that are sign-changing solutions thanks to u-≤z≤u+ and the extremality of the solutions u-, u+ obtained in Theorem 3.10 which proves the assertion.
Therefore, it remains to prove the existence of sign-changing solutions under the assumptions that the global minimizerv0 of E0 is equal to u+, and u- is a strict local minimizer of E0. This implies the existence of ρ∈(0,∥u--u+∥) such that(3.57)E0(u+)≤E0(u-)<inf{E0(u):u∈∂Bρ(u-)},
where ∂Bρ(u-)={u∈V0:∥u-u-∥=ρ}. The functional E0 satisfies the Palais-Smale condition, because it is bounded below, locally Lipschitz continuous, and coercive; see, for example, [40, Corollary 2.4]. Thus in view of (3.57) we may apply the nonsmooth version of Ambrosetti-Rabinowitz's Mountain-Pass Theorem (see, e.g., [41, Theorem 3.4]) which ensures the existence of a critical point u0∈V0 satisfying 0∈∂E0(u0) and
(3.58)inf{E0(u):u∈∂Bρ(u-)}≤E0(u0)=infγ∈Γmaxt∈[-1,1]E0(γ(t)),
where
(3.59)Γ={γ∈C([-1,1],V0):γ(-1)=u-,γ(1)=u+}.
It is clear from (3.57) and (3.58) that u0≠u-and u0≠u+, and thus u0 is a sign-changing solution provided u0≠0. To prove the latter we claim
(3.60)E0(u0)<0
for which it suffices to construct a path γ̂∈Γ such that
(3.61)E0(γ̂(t))<0∀t∈[-1,1].
The construction of such a path γ̂ can be done by adopting an approach due to the authors in [3] and applying the Second Deformation Lemma for locally Lipschitz functionals as it can be found in [42, Theorem 2.10]. This completes the proof.Remark 3.15.
The multiplicity results and the existence of sign-changing solutions obtained in this section generalize very recent results due to the authors obtained in [3, 4, 7]. Moreover, if the function t↦g(x,t) is continuous on ℝ and a=b=λ, then ∂G(x,ξ)={g(x,ξ)}, and problem (1.7) reduces to
(3.62)u∈V0:-Δpu=λ|u|p-2u-g(x,u)inV0*.
However, even in this setting the results obtained here are more general than obtained in [6, Theorem 3.9], because we do not assume that g(x,t)t≥0 for all t∈ℝ.Remark 3.16.
Theorem3.14 improves also Corollary 3.2 of [8]. In fact, let p=2, let u∈V0 be a solution of (1.7) in case a=b=λ and g(x,t)≡g(t), (x,t)∈Ω×ℝ with η∈Lq′(Ω) satisfying η(x)∈∂G(u(x)). By definition of Clarke's gradient we have, for any φ∈V0,
(3.63)η(x)φ(x)≤G0(u(x);φ(x))a.e.inΩ.
As u is a solution, the following holds: u∈V0 and (p=2),
(3.64)∫Ω∇u∇φdx=λ∫Ωuφdx-∫Ωηφdx,
which yields
(3.65)-∫Ω∇u∇φdx+λ∫Ωuφdx=∫Ωηφdx≤∫ΩG0(u;φ)dx,∀φ∈V0,
That is, u turns out to be a solution of the hemivariational inequality studied in [8]. Since the hypotheses of [8, Corollary 3.2] imply (g1)–(g4), the assertion follows.Remark 3.17.
Multiplicity results for a nonsmooth version of problem (1.4) in form of (1.2) can be established under structure conditions for Clarke's gradient ∂j similar to H(f).
Multiplicity and sign-changing solutions results have been obtained recently in [43] for the following nonlinear Neumann-type boundary value problem: find u∈V∖{0} and parameters a,b∈ℝ such that(3.66)-Δpu=f(x,u)-|u|p-2uinΩ,|∇u|p-2∂u∂ν=a(u+)p-1-b(u-)p-1+g(x,u)on∂Ω.
For problem (3.66) conditions on the parameters have been given in terms of the “Steklov-Fučik” spectrum to ensure multiplicity results.



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