# Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities

Authors: Leszek Gasiński; Nikolaos S. Papageorgiou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a parametric semilinear Dirichlet problem with an unbounded and indefinite potential. In the reaction we have the competing effects of a sublinear (concave) term and of a superlinear (convex) term. Using variational methods coupled with suitable truncation techniques, we prove two multiplicity theorems for small values of the parameter. Both theorems produce five nontrivial smooth solutions, and in the second theorem we provide precise sign information for all the solutions.


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## Body

## 1. Introduction

LetΩ∈ℝN be a bounded domain with a C2-boundary ∂Ω. In this paper we study the following parametric nonlinear Dirichlet problem:
((P)λ)-Δu(z)+β(z)u(z)=λg(z,u(z))+f(z,u(z))inΩ,u|∂Ω=0,λ>0.
Here β∈Ls(Ω) with s>N/2(N⩾2) is a potential function which may change sign (indefinite potential). Also λ>0 is a parameter, and g,f:Ω×ℝ→ℝ are Carathéodory functions (i.e., for all ζ∈ℝ, functions z↦g(z,ζ) and z↦f(z,ζ) are measurable and for almost all z∈Ω, functions ζ↦g(z,ζ) and ζ↦f(z,ζ) are continuous). We assume that for almost all z∈Ω, the function ζ↦g(z,ζ) is strictly sublinear near ±∞, while the function ζ↦f(z,ζ) is superlinear near ±∞. So, problem ((P)λ) exhibits competing nonlinearities of the concave-convex type. This situation was first studied with β≡0 and the right hand side nonlinearity being
(1.1)λ|ζ|q-2ζ+|ζ|r-2ζ,
with
(1.2)1<q<2<r<2*={2NN-2ifN>2,+∞ifN=2,
by Ambrosetti et al. [1]. In [1], the authors focus on positive solutions and proved certain bifurcation-type phenomena as λ>0 varies. Further results in this direction can be found in the works of I'lyasov [2], Li et al. [3], Lubyshev [4], and Rădulescu and  Repovš [5]. In all the aforementioned works β≡0. We should also mention the recent work of Motreanu et al. [6], where the authors consider equations driven by the p-Laplacian, with concave term of the form |ζ|q-2ζ (where 1<q<p) and a perturbation exhibiting an asymmetric behaviour at +∞ and at -∞ ((p-1)-superlinear near +∞ and (p-1)-sublinear near -∞). They prove multiplicity results producing four nontrivial solutions with sign information.In this work, we prove two multiplicity results for problem ((P)λ) when the parameter λ>0 is small. In both results we produce five nontrivial smooth solutions, and in the second we provide precise sign information for all the solutions. For the superlinear (“convex”) nonlinearity f(z,·), we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Instead, we use a more general condition, which incorporates in our framework also superlinear perturbations with “slow” growth near ±∞, which do not satisfy the Ambrosetti-Rabinowitz condition. We should point out that none of the works mentioned earlier provide sign information for all the solutions (in particular, none of them produced a nodal (sign changing) solution) and all use the Ambrosetti-Rabinowitz condition to express the superlinearity of the “convex” contribution in the reaction.Our approach is variational based on the critical point theory which is combined with suitable truncation techniques. In the next section we recall the main mathematical tools we will use in the analysis of problem ((P)λ). We also introduce the hypotheses on the terms g and f.


## 2. Mathematical Background and Hypotheses

LetX be a Banach space and let X* be its topological dual. By 〈·,·〉 we denote the duality brackets for the pair (X*,X). Let φ∈C1(X). We say that φ satisfies the Cerami condition, if the following is true.Every sequence{xn}n⩾1⊆X, such that {φ(xn)}n⩾1⊆ℝ is bounded and
(2.1)(1+‖xn‖)φ′(xn)⟶0inX*,
admits a strongly convergent subsequence.This compactness-type condition is in general weaker than the usual Palais-Smale condition. However, the Cerami condition suffices to prove a deformation theorem and from it we derive the minimax theory for certain critical values ofφ∈C1(X) (see Gasiński and Papageorgiou [7] and Motreanu and Rădulescu [8]). In particular, we can have the following theorem, known in the literature as the mountain pass theorem.Theorem 2.1.
IfX is a Banach space, φ∈C1(X) satisfies the Cerami condition, x0,x1∈X,  ϱ>0∥x1-x0∥>ϱ:
(2.2)max⁡⁡{φ(x0),φ(x1)}<inf⁡⁡{φ(x):‖x-x0‖=ϱ}=ηϱ,c=inf⁡γ∈Γmax⁡t∈[0,1]φ(γ(t)),
where
(2.3)Γ={γ∈C([0,1];X):γ(0)=x0,γ(1)=x1}.
Then c⩾ηϱ and c are a critical value of φ.In the analysis of problem ((P)λ), in addition to the Sobolev space H01(Ω), we will also use the Banach space
(2.4)C01(Ω¯)={u∈C1(Ω¯):u|∂Ω=0}.
This is an ordered Banach space with positive cone:
(2.5)C+={u∈C01(Ω¯):u(z)⩾0∀z∈Ω¯}.
This cone has a nonempty interior given by
(2.6)int⁡C+={u∈C+:u(z)>0∀z∈Ω,∂u∂n(z)<0∀z∈∂Ω},
where n(·) is the outward unit normal on ∂Ω.In the proof of the second multiplicity theorem and in order to produce a nodal (sign changing) solution, we will also use critical groups. So, let us recall their definition. Letφ∈C1(X) and let c∈ℝ. We introduce the following sets:
(2.7)φc={x∈X:φ(x)⩽c},Kφ={x∈X:φ′(x)=0}.
Also, if (Y1,Y2) is a topological pair with Y2⊆Y1⊆X and k⩾0 is an integer, by Hk(Y1,Y2) we denote the kth relative singular homology group for the pair (Y2,Y1) with integer coefficients. The critical groups of φ at an isolated critical point x0∈X of φ with φ(x0)=c are defined by
(2.8)Ck(φ,x0)=Hk(φc∩U,φc∩U∖{x0})∀k⩾0,
where U is a neighbourhood of x0, such that Kφ∩φc∩U={x0}. The excision property of singular homology implies that this definition is independent of the particular choice of the neighbourhood U.Using the spectral theorem for compact self-adjoint operators, we can show that the differential operator(2.9)H01(Ω)∋u⟼-Δu+βu
has a sequence of distinct eigenvalues {λ̂k}k⩾1, such that
(2.10)λ̂k⟶+∞ask⟶+∞.
The first eigenvalue is simple and admits the following variational characterization:
(2.11)λ̂1=inf⁡⁡{σ(u)‖u‖22:u∈H01(Ω),u≠0},
where
(2.12)σ(u)=‖∇u‖22+∫Ωβ(z)u(z)2dz∀u∈H01(Ω).
Moreover, the corresponding eigenfunction û∈C01(Ω¯) does not change sign, and in fact we can take û(z)>0 for all z∈Ω (see Gasiński  and Papageorgiou [9]). Using (2.11) and this property of the principal eigenfunction, we can have the following lemma (see Gasiński and Papageorgiou [9, Lemma 2.1]).Lemma 2.2.
Ifη∈Ls(Ω), η(z)⩽λ̂1 for almost all z∈Ω, η≠λ̂1, then there exists ĉ>0, such that
(2.13)σ(u)-∫Ωηu2dz⩾ĉ‖u‖2∀u∈H01(Ω).Also, from Gasiński and Papageorgiou [9] we know that there exist ĉ0,ĉ1>0, such that(2.14)‖u‖2⩽ĉ0(σ(u)+ĉ1‖u‖22)∀u∈H01(Ω),
with ∥u∥=∥∇u∥2, the norm of the Sobolev space H01(Ω).Next we state the hypotheses on the two componentsg and f of the reaction in problem ((P)λ).Let(2.15)G(z,ζ)=∫0ζg(z,s)ds,F(z,ζ)=∫0ζf(z,s)ds.Hg̲:
g:Ω×ℝ→ℝ is a Carathéodory function, such that g(z,0)=0 for almost all z∈Ω.(i)
For everyϱ>0, there exists a function aϱ∈L∞(Ω)+, such that
(2.16)|g(z,ζ)|⩽aϱ(z)foralmostallz∈Ω,all|ζ|⩽ϱ.(ii)
We have(2.17)lim⁡ζ→±∞g(z,ζ)ζ=0uniformlyforalmostallz∈Ω.(iii)
There exist constantsc1,c2>0, 1<q<μ<2 and δ0>0, such that
(2.18)c1|ζ|q⩽g(z,ζ)ζforalmostallz∈Ω,allζ∈R,μG(z,ζ)-g(z,ζ)ζ⩾c2|ζ|qforalmostallz∈Ω,all|ζ|⩽δ0.(iv)
For everyϱ>0, we can find γϱ>0, such that for almost all z∈Ω(2.19)themap[-ϱ,ϱ]∋ζ⟼g(z,ζ)+γϱ|ζ|q-2ζisnondecreasing.Remark 2.3.
HypothesisHg(ii) implies that for almost all z∈Ω, the function g(z,·) is strictly sublinear near ±∞. Hence g(z,·) is the “concave” component in the reaction of ((P)λ) (the terminology “concave” and “convex” nonlinearities is due to Ambrosetti [1]). Note that hypothesis Hg(iii) implies that for almost all z∈Ω, the function g(z,·) has a similar growth near 0 that is, we have a concave term near zero. Hypothesis Hg(iv) is weaker than assuming the monotonicity of g(z,·) for almost all z∈Ω.Hf̲:
f:Ω×ℝ→ℝ is a Carathéodory function, such that f(z,0)=0 for almost all z∈Ω.(i)
There exista∈L∞(Ω)+, c>0 and r∈(2,2*), such that
(2.20)|f(z,ζ)|⩽a(z)+c|ζ|r-1foralmostallz∈Ω,allζ∈R.(ii)
We have(2.21)lim⁡ζ→±∞F(z,ζ)ζ2=+∞uniformlyforalmostallz∈Ω.(iii)
There exist functionsη0,η̂0∈L∞(Ω)+, such that η0(z)⩽λ̂1 for almost all z∈Ω, η0≠λ̂1 and
(2.22)η̂0(z)⩽lim⁡inf⁡ζ→0f(z,ζ)ζ⩽limsup⁡ζ→0f(z,ζ)ζ⩽η0(z)
uniformly for almost all z∈Ω.(iv)
For everyϱ>0, we can find γ̂ϱ>0, such that for almost all z∈Ω, we have(2.23)themap[-ϱ,ϱ]∋ζ⟼f(z,ζ)+γ̂ϱζisnondecreasing.Remark 2.4.
HypothesisHf(ii) implies that for almost all z∈Ω, the function F(z,·) is superquadratic near ±∞. Evidently, this is satisfied if the function f(z,·) is superlinear near ±∞, that is, when
(2.24)lim⁡ζ→±∞f(z,ζ)ζ=+∞uniformlyforalmostallz∈Ω.
So, f(z,ζ) is the “convex” component of the reaction which “competes” with the “concave” component g(z,ζ).Note that inHf, we did not include the Ambrosetti-Rabinowitz condition to characterize the superlinearity of f(z,·). We recall that the Ambrosetti-Rabinowitz condition says that there exist τ>2 and M>0, such that(2.25)0<τF(z,ζ)⩽f(z,ζ)ζuniformlyforalmostallz∈Ω,all|ζ|⩾M,(2.26)essinf⁡ΩF(⋅,M)>0.
Integrating (2.25) and using (2.26), we obtain the weaker condition
(2.27)c3|ζ|τ⩽F(z,ζ)uniformlyforalmostallz∈Ω,all|ζ|⩾M.
Therefore, for almost all z∈Ω, the function
(2.28)ζ⟼λG(z,ζ)+F(z,ζ)
is superquadratic with at least τ-growth near ±∞. Hence the Ambrosetti-Rabinowitz condition excludes superlinear perturbations with “slower” growth near ±∞. For this reason, here we employ a weaker condition. So, let
(2.29)ξλ(z,ζ)=(λg(z,ζ)+f(z,ζ))ζ-2(λG(z,ζ)+F(z,ζ)).We employ the following hypothesis.H0̲:
For everyλ>0, there exists a function ϑλ*∈L1(Ω)+, such that
(2.30)ξλ(z,ζ)⩽ξλ(z,y)+ϑλ*(z)foralmostallz∈Ω,all0⩽ζ⩽y,ξλ(z,ζ)⩽ξλ(z,y)+ϑλ*(z)foralmostallz∈Ω,ally⩽ζ⩽0.Remark 2.5.
HypothesisH0 is a generalized version of a condition first introduced by Li and Yang [10], where the reader can find other possible extensions of the Ambrosetti-Rabinowitz condition and comparisons between them. Hypothesis H0 is a quasimonotonicity condition on ξλ(z,·), and it is satisfied if there exists M>0, such that for almost all z∈Ω, the function
(2.31)ζ⟼λg(z,ζ)+f(z,ζ)ζ
is increasing on [M,+∞] and is decreasing on [-∞,-M] (see Li and Yang [10]).Example 2.6.
The following pairs of functions satisfy hypothesesHg, Hf, and H0 (for the sake of simplicity we drop the z-dependence):
(2.32)g1(ζ)=|ζ|q-2ζ,f1(ζ)=|ζ|p-2ζ+η0ζ,g2(ζ)=|ζ|q-2ζ,f2(ζ)=|ζ|p-2ζln⁡⁡(1+|ζ|)+η0ζ,g3(ζ)=|ζ|q-2ζ-|ζ|τ-2ζ,f3(ζ)={η0(ζ-|ζ|r-2ζ)if|ζ|⩽1,|ζ|p-2ζln⁡⁡|ζ|if|ζ|>1,
with 1<q<τ<2<p<2*, 2<r, η0<λ̂1.
Note thatf2 and f3 do not satisfy the Ambrosetti-Rabinowitz condition (see (2.25)-(2.26)).For everyu∈H01(Ω), we set
(2.33)‖u‖=‖∇u‖2
(by virtue of the Poincaré inequality). We mention that the notation ∥·∥ will be also used to denote the ℝN-norm. It will always be clear from the context which norm is used. For ζ∈ℝ, let ζ±=max{±ζ,0}. Then for u∈W01,p(Ω), we set u±(·)=u(·)±. We have u±∈H01(Ω), |u|=u++u-, and u=u+-u-. For a given measurable function h:Ω×ℝ→ℝ (e.g., a Carathéodory function), we set(2.34)Nh(u)(⋅)=h(⋅,u(⋅))∀u∈W01,p(Ω).
Finally, let A∈ℒ(H01(Ω),H-1(Ω)) be the operator, defined by
(2.35)〈A(u),y〉=∫Ω(∇u,∇y)RNdz∀u,y∈H01(Ω).
For the properties of the operator A we refer to Gasiński and Papageorgiou [11, Proposition 3.1, page 852].


## 3. Solutions of Constant Sign

In this section, forλ>0 small, we generate four nontrivial smooth solutions of constant sign (two positive and two negative). To this end, we introduce the following modifications of the nonlinearities g(z,·) and f(z,·):(3.1)g±(z,ζ)=g(z,±ζ±),f̂±(z,ζ)=f(z,±ζ±)+ĉ1(±ζ±),
with ĉ1>0 as in (2.14). These modifications are Carathéodory functions. We set(3.2)G±(z,ζ)=∫0ζg±(z,s)ds,F̂±(z,ζ)=∫0ζf̂±(z,s)ds
and consider the C1-functionals φ̂λ±:H01(Ω)→ℝ, defined by(3.3)φ̂λ±(u)=12σ(u)+ĉ12‖u‖22-λ∫ΩG±(z,u(z))-∫ΩF̂±(z,u(z))dz∀u∈H01(Ω).Proposition 3.1.
If hypothesesHg, Hf, and H0 hold and λ>0, then the functionals φ̂λ± satisfy the Cerami condition.Proof.
We do the proof forφ̂λ+, the proof for φ̂λ- being similar.
So, let{un}n⩾1⊆H01(Ω) be a sequence, such that
(3.4)|φ̂λ+(un)|⩽M1∀n⩾1,
for some M1>0 and
(3.5)(1+‖un‖)(φ̂λ+)′(un)⟶0inH-1(Ω).
From (3.5), we have
(3.6)|〈A(un),h〉+∫Ω(β+ĉ1)unhdz-λ∫Ωg+(z,un)hdz-∫Ωf̂+(z,un)hdz|⩽ɛn‖h‖1+‖un‖∀h∈H01(Ω),
with ɛn↘0. In (3.6) we choose h=-un-∈H01(Ω). Then
(3.7)|σ(un-)+ĉ1‖un-‖22|⩽ɛn∀n⩾1,
so
(3.8)1ĉ0‖un-‖2⩽ɛn∀n⩾1
(see (2.14)), and hence
(3.9)un-⟶0inH01(Ω).
From (3.4) and (3.9), we have
(3.10)σ(un+)-λ∫Ω2G+(z,un+)dz-∫Ω2F(z,un+)dz⩽M2∀n⩾1,
for some M2>0. Also, if in (3.6) we choose h=un+∈H01(Ω), then
(3.11)-σ(un+)+λ∫Ωg(z,un+)un+dz+∫Ωf(z,un+)un+dz⩽ɛn∀n⩾1.
Adding (3.10) and (3.11), we obtain
(3.12)∫Ωξλ(z,un+)dz≤M3∀n⩾1,
for some M3>0 (see (2.29) for the definition of ξλ).
Claim 1.
The sequence{un+}n⩾1⊆H01(Ω) is bounded.
Arguing by contradiction, suppose that the claim is not true. Then by passing to a subsequence if necessary, we may assume that(3.13)‖un+‖⟶+∞.
Let
(3.14)yn=un+‖un+‖∀n⩾1.
Then
(3.15)‖yn‖=1∀n⩾1.
And so, passing to a subsequence if necessary, we may assume that
(3.16)yn⟶yweaklyinH01(Ω),(3.17)yn⟶yinLr(Ω).
If y≠0, then
(3.18)un+(z)⟶+∞foralmostallz∈Ω+={y>0}
(recall that y⩾0). So, by virtue of hypotheses Hg(ii) and Hf(ii), for almost all z∈Ω+, we have
(3.19)lim⁡n→+∞G(z,un+(z))‖un+‖2=lim⁡n→+∞G(z,un+(z))un+(z)2yn(z)2=0,lim⁡n→+∞F(z,un+(z))‖un+‖2=lim⁡n→+∞F(z,un+(z))un+(z)2yn(z)2=+∞.
Then Fatou's lemma implies that
(3.20)lim⁡n→+∞(λ∫ΩG(z,un+)‖un+‖2dz+∫ΩF(z,un+)‖un+‖2dz)=+∞.
But from (3.4) and (3.9), we have
(3.21)λ∫ΩG(z,un+)dz+∫ΩF(z,un+)dz⩽M4+|σ(un+)|∀n⩾1,
for some M4>0, so
(3.22)λ∫ΩG(z,un+)‖un+‖2dz+∫ΩF(z,un+)‖un+‖2dz⩽M5∀n⩾1,
for some M5>0 (since the sequence {σ(yn)}n⩾1 is bounded in ℝ). Comparing (3.20) and (3.22), we reach a contradiction.
So, we havey=0. We fix μ>0 and set
(3.23)vn=(2μ)1/2yn∀n⩾1.
Evidently
(3.24)vn⟶0inLr(Ω)
(see (3.17)). Hence by Krasnoselskii's theorem (see Gasiński and Papageorgiou [12, Proposition 1.4.14, page 87] and hypotheses Hg(i) and Hf(i)), we have
(3.25)∫ΩG(z,vn(z))dz⟶0,∫ΩF(z,vn(z))dz⟶0.
Since ∥un+∥→+∞, we can find n0⩾1, such that
(3.26)0<(2μ)1/21‖un+‖<1∀n⩾n0.
Let tn∈[0,1] be such that
(3.27)φ̂λ+(tnun+)=max⁡0⩽t⩽1φ̂λ+(tun+).
By virtue of (3.26), we have
(3.28)φ̂λ+(tnun+)⩾φ̂λ+(vn)=2μσ(yn)-λ∫ΩG(z,vn)dz-∫ΩF(z,vn)dz=2μ+2μ∫Ωβyn2dz-λ∫ΩG(z,vn)dz-∫ΩF(z,vn)dz.
Note that
(3.29)∫Ωβyn2dz⟶0.
This fact together with (3.25) and (3.28) implies that
(3.30)φ̂λ+(tnun+)⩾μ∀n⩾n1,
for some n1⩾n0. Since μ>0 is arbitrary, we infer that
(3.31)φ̂λ+(tnun+)⟶+∞.
Note that
(3.32)φ̂λ+(0)=0,φ̂λ+(un+)⩽M6∀n⩾1,
for some M6>0 (see (3.4) and (3.9)).
Hence (3.31) implies that there exists n2⩾n1, such that
(3.33)tn∈(0,1)∀n⩾n2.
And so from the choice of tn, we have
(3.34)ddtφ̂λ+(tun+)|t=tn=0∀n⩾n2,
so
(3.35)〈(φ̂λ+)′(tnun),un〉=0∀n⩾n2,
thus
(3.36)〈(φ̂λ+)′(tnun),tnun〉=0∀n⩾n2,
and hence
(3.37)σ(tnun+)=λ∫Ωg(z,tnun+)tnun+dz+∫Ωf(z,tnun+)tnun+dz.
Hypothesis H0 implies that
(3.38)∫Ωξλ(z,tnun+)dz⩽∫Ωξλ(z,un+)dz+‖ϑλ*‖1∀n⩾1,
so
(3.39)2φ̂λ+(tnun+)⩽∫Ωξλ(z,un+)dz+‖ϑλ*‖1∀n⩾1
(see (3.37)), and thus
(3.40)2φ̂λ+(tnun+)⩽M3+‖ϑλ*‖1∀n⩾1
(see (3.12)).
Comparing (3.31) and (3.40), we reach a contradiction. This proves the claim.
By virtue of the claim and (3.9), we have that the sequence {un}n⩾1⊆H01(Ω) is bounded. So, we may assume that
(3.41)un⟶uweaklyinH01(Ω),(3.42)un⟶uinLr(Ω).
In (3.6) we choose h=un-u, pass to the limit as n→+∞, and use (3.42). Then
(3.43)lim⁡n→+∞〈A(un),un-u〉=0,
so
(3.44)‖un‖=‖∇un‖2⟶‖∇u‖2=‖u‖
(see Gasiński and Papageorgiou [11, Proposition 3.1, page 852]), and thus
(3.45)un⟶uinH01(Ω)
(by the Kadec-Klee property of Hilbert spaces). This proves that φ̂λ+ satisfies the Cerami condition.
Similarly we show thatφ̂λ- also satisfies the Cerami condition.Our aim is to apply Theorem2.1 (the mountain pass theorem) to two functionals φ̂λ+ and φ̂λ-. We have checked that both functionals satisfy the Cerami condition. So, it remains to show that they satisfy the mountain pass geometry as it is described in Theorem 2.1.The next proposition is a crucial step in satisfying the mountain pass geometry for the two functionalsφ̂λ+ and φ̂λ-.Proposition 3.2.
If hypothesesHg and Hf hold, then there exist λ±*>0, such that for every λ∈(0,λ±*), we can find ϱλ±>0, such that
(3.46)inf⁡⁡{φ̂λ±(u):‖u‖=ϱλ±}=η̂λ±>0.Proof.
HypothesesHg(i) and (ii) imply that for a given ɛ>0, we can find c4=c4(ɛ)>0, such that
(3.47)g(z,ζ)⩽ɛζ+c4ζq-1foralmostallz∈Ω,allζ⩾0,
so
(3.48)G(z,ζ)⩽ɛ2ζ2+c4qζqforalmostallz∈Ω,allζ⩾0.
Similarly hypotheses Hf(i) and (iii) imply that for a given ɛ>0, we can find c5=c5(ɛ)>0, such that
(3.49)f(z,ζ)⩽(η0(z)+ɛ)ζ+c5ζr-1foralmostallz∈Ω,allζ⩾0,
so
(3.50)F(z,ζ)⩽12(η0(z)+ɛ)ζ2+c5rζrforalmostallz∈Ω,allζ⩾0.
Then, for u∈H01(Ω), we have
(3.51)φ̂λ+(u)=12σ(u)+ĉ12‖u‖22-λ∫ΩG+(z,u)dz-∫ΩF̂+(z,u)dz⩾12σ(u+)-12∫Ωη0(u+)2dz-ɛ2‖u+‖22-c5r‖u+‖rr-λɛ2‖u+‖22-λc4q‖u+‖qq+12σ(u-)+ĉ12‖u-‖22⩾12(ĉ-ɛ(λ+1)λ̂1)‖u+‖2+12ĉ0‖u-‖2-c6(‖u‖r+λ‖u‖q),
for some c6>0 (see (3.48), (3.50), (2.11), (2.14), and Lemma 2.2).
Choosingɛ∈(0,λ̂1ĉ/(λ+1)), we have
(3.52)φ̂λ+(u)⩾c7‖u‖2-c6(‖u‖r+λ‖u‖q)=(c7-c6(‖u‖r-2+λ‖u‖q-2))‖u‖2,
for some c7>0. Let
(3.53)μλ(t)=tr-2+λtq-2∀t>0.
Since q<2<r, we have
(3.54)lim⁡t→0+μλ(t)=lim⁡r→+∞μλ(t)=+∞.
Also μλ is continuous in (0,+∞). Therefore, we can find t0∈(0,+∞), such that
(3.55)μλ(t0)=inf⁡t>0μλ(t),
so
(3.56)μλ′(t0)=0,
and thus
(3.57)t0=t0(λ)=(λ(2-q)r-2)1/(r-q).
Evidently
(3.58)μλ(t0)⟶0asλ⟶0+.
Hence we can find λ+*>0, such that
(3.59)μλ(t0)<c7c6∀λ∈(0,λ+*),
so
(3.60)inf⁡⁡{φ̂λ+(u):‖u‖=ϱλ+=t0(λ)}=η̂λ+>0
(see (3.52)).
Similarly, forφ̂λ-, we can find λ-*>0, such that for all λ∈(0,λ-*) there exists ϱλ->0, such that
(3.61)inf⁡⁡{φ̂λ-(u):‖u‖=ϱλ-}=η̂λ->0.With the next proposition we complete the mountain pass geometry for problem ((P)λ).Proposition 3.3.
If hypothesesHg and Hf hold, λ>0 and ũ∈intC+ with ∥ũ∥2=1, then
(3.62)φ̂λ±(tũ)⟶-∞ast⟶±∞.Proof.
By virtue of hypothesesHg(i) and (ii), for a given ɛ>0, we can find c8=c8(ɛ)>0, such that
(3.63)G(z,ζ)⩾-ɛ2ζ2-c8,foralmostallz∈Ω,allζ∈R.
Similarly, hypotheses Hf(i) and (ii) imply that for any given ξ>0, we can find c9=c9(ξ)>0, such that
(3.64)F(z,ζ)⩾ξ2ζ2-c9,foralmostallz∈Ω,allζ∈R.
Then, we have
(3.65)φ̂λ+(tũ)=t22σ(ũ)+(λɛ-ξ2)t2+c10⩽t22(‖ũ‖2+‖β‖s‖ũ‖2s′+λɛ-ξ)+c10
for some c10>0 (see (3.63), (3.64) and recall that ũ∈intC+, ∥ũ∥2=1 and 1/s+1/s′=1).
Sinceξ>0 is arbitrary, choosing
(3.66)ξ>λɛ+‖ũ‖2+‖β‖s‖ũ‖2s′,
from (3.65), we infer that
(3.67)φ̂λ+(tũ)⟶-∞ast⟶±∞.Now we are ready to produce the first two nontrivial smooth solutions of constant sign. In what follows, we set(3.68)λ*=min⁡⁡{λ-*,λ+*}.Proposition 3.4.
If hypothesesHg, Hf, and H0 hold and λ∈(0,λ*], then problem ((P)λ) has at least two nontrivial smooth solutions of constant sign:
(3.69)u0,v0∈C01(Ω¯),withv0(z)<0<u0(z)∀z∈Ω.Proof.
Propositions3.1, 3.2, and 3.3 permit the application of the mountain pass theorem (see Theorem 2.1) for the functional φ̂λ+, and so we obtain u0∈H01(Ω), such that
(3.70)φ̂λ+(0)=0<η̂λ+⩽φ̂λ+(u0),(3.71)(φ̂λ+)′(u0)=0.
From (3.70), we see that u0≠0. From (3.71), we have
(3.72)A(u0)+βu0+ĉ1u0=λNg+(u0)+Nf̂+(u0).
On (3.72) we act with -u0-∈H01(Ω) and obtain
(3.73)σ(u0-)+ĉ1‖u0-‖22=0,
so
(3.74)1ĉ0‖u0-‖2⩽0
(see (2.14)); hence u0⩾0, u0≠0.
Therefore (3.72) becomes
(3.75)A(u0)+βu0=λNg(u0)+Nf(u0),
so
(3.76)-Δu0(z)+β(z)u0(z)=λg(z,u0(z))+f(z,u0(z))inΩ,u0|∂Ω=0.
From the regularity theory for Dirichlet problems (see Struwe [13, pp. 217–219]), we have that u0∈C01(Ω¯). Moreover, invoking the weak Harnack inequality of Pucci and Serrin [14, page 154], we have that u0(z)>0 for all z∈Ω.
Similarly working withφ̂λ-, this time we obtain a nontrivial smooth negative solution v0∈C01(Ω¯) with v0(z)<0 for all z∈Ω.We can improve the conclusion of this proposition by strengthening the condition on the potentialβ:Hβ̲:
β∈Ls(Ω) with s>N/2 and β+∈L∞(Ω)+.Remark 3.5.
So, the potential function is bounded from above but in general can be unbounded from below.Proposition 3.6.
If hypothesesHg, Hf, H0, and Hβ hold and λ∈(0,λ*), then problem ((P)λ) has at least two nontrivial smooth solutions of constant sign:
(3.77)u0∈int⁡C+,v0∈-int⁡C+.Proof.
From Proposition3.4, we already have two solutions:
(3.78)u0,v0∈C01(Ω¯),withv0(z)<0<u0(z)∀z∈Ω.
We have
(3.79)-Δu0(z)+β(z)u0(z)=λg(z,u0(z))+f(z,u0(z))⩾f(z,u0(z))foralmostallz∈Ω
(see hypothesis Hg(iii)). Let ϱ=∥u0∥∞ and let γ̂ϱ>0 be as postulated by hypothesis Hf(iv). Then from (3.79), we have
(3.80)-Δu0(z)+(β(z)+γ̂ϱ)u0(z)⩾f(z,u0(z))+γ̂ϱu0(z),foralmostallz∈Ω,
so
(3.81)Δu0(z)⩽(‖β+‖∞+γ̂ϱ)u0(z)foralmostallz∈Ω.
and thus u0∈intC+ (see Vázquez [15] and Pucci and Serrin [14, page 120]).
Similarly for the negative solutionv0.To continue and produce additional nontrivial smooth solutions of constant sign, we need to keep hypothesesHβ.Proposition 3.7.
If hypothesesHg, Hf, H0 and Hβ hold and λ∈(0,λ*), then problem ((P)λ) has at least four nontrivial smooth solutions of constant sign:
(3.82)u0,û∈int⁡C+,û-u0∈int⁡C+,v0,v̂∈-int⁡C+,v0-v̂∈int⁡C+.Proof.
From Proposition3.6, we already have two solutions:
(3.83)u0∈int⁡C+,v0∈-int⁡C+.
We introduce the following truncation perturbation of the reaction of the problem ((P)λ):
(3.84)hλ+(z,ζ)={λg(z,u0(z))+f(z,u0(z))+ĉ1u0(z)ifζ⩽u0(z),λg(z,ζ)+f(z,ζ)+ĉ1ζifu0(z)<ζ.
This is a Carathéodory function. We set
(3.85)Hλ+(z,ζ)=∫0ζhλ+(z,s)ds
and consider the C1-functional ψλ+:H01(Ω)→ℝ, defined by
(3.86)ψλ+(u)=12σ(u)+ĉ12‖u‖22-∫ΩHλ+(z,u(z))dz∀u∈H01(Ω).Claim 2.
We haveKψλ+⊆[u0), where
(3.87)[u0)={u∈H01(Ω):u0(z)⩽u(z)foralmostallz∈Ω}.
Letũ∈Kψλ+. Then(3.88)A(ũ)+(β+ĉ1)ũ=Nhλ+(ũ).
On (3.88) we act with (u0-ũ)+∈H01(Ω). Then
(3.89)〈A(ũ),(u0-ũ)+〉+∫Ω(β+ĉ1)ũ(u0-ũ)+dz=∫Ωhλ+(z,ũ)(u0-ũ)+dz=∫Ω(λg(z,u0)+f(z,u0)+ĉ1u0)(u0-ũ)+dz=〈A(u0),(u0-û)+〉+∫Ω(β+ĉ1)u0(u0-ũ)+dz
(see (3.84)), so
(3.90)〈A(u0-ũ),(u0-ũ)+〉+∫Ωβ(u0-ũ)(u0-ũ)+dz+ĉ1‖(u0-ũ)+‖22=0,
thus
(3.91)σ((u0-ũ)+)+ĉ1‖(u0-ũ)+‖22=0,
hence
(3.92)1ĉ0‖(u0-ũ)+‖2⩽0
(see (2.14)), and so finally u0⩽ũ. This proves Claim 2.Claim 3.
We may assume thatu0 is a local minimizer of ψλ+.
Letμ∈(λ,λ*), and consider problem (P)μ. As we did in the proof of Proposition 3.4, via the mountain pass theorem, we obtain a nontrivial smooth positive solution uμ, and by virtue of the strong maximum principle, we have uμ∈intC+ (see the proof of Proposition 3.7). Then
(3.93)-Δuμ(z)+β(z)uμ(z)=μg(z,uμ(z))+f(z,uμ(z))⩾λg(z,uμ(z))+f(z,uμ(z))foralmostallz∈Ω
(see Hg(iii) and recall that λ<μ).
We consider the following truncation perturbation of the reaction of problem ((P)λ):
(3.94)γλ+(z,ζ)={0ifζ<0,λg(z,ζ)+f(z,ζ)+ĉ1ζif0⩽ζ⩽uμ(z),λg(z,uμ(z))+f(z,uμ(z))+ĉ1uμ(z)ifuμ(z)<ζ.
This is a Carathéodory function. We set
(3.95)Γλ+(z,ζ)=∫0ζγλ+(z,s)ds
and consider the C1-functional ψ̂λ+:H01(Ω)→ℝ, defined by
(3.96)ψ̂λ+(u)=12σ(u)+ĉ12‖u‖22-∫ΩΓλ+(z,u(z))dz∀u∈H01(Ω).
From (3.94), it is clear that ψ̂λ+ is coercive. Also, using the Sobolev embedding theorem, we check that ψ̂λ+ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find uλ∈H01(Ω), such that
(3.97)ψ̂λ+(uλ)=inf⁡u∈H01(Ω)ψ̂λ+(u).
By virtue of hypothesis Hf(iii), we can find c11>0 and δ̂0>0, such that
(3.98)F(z,ζ)⩾-c112ζ2foralmostallz∈Ω,all|ζ|⩽δ̂0.
Let ũ∈intC+ and let t∈(0,1) be small, such that
(3.99)tũ⩽uμ,tũ(z)⩽δ̂0∀z∈Ω¯
(recall that uμ∈intC+). Then, we have
(3.100)ψ̂λ+(tũ)=t22σ(ũ)+t2ĉ12‖ũ-‖22-λ∫ΩG(z,tũ)dz-∫ΩF(z,tũ)dz⩽t22(σ(ũ)+(ĉ1+c11)‖ũ‖22)-λtqc1q‖ũ‖qq
(see (3.94), hypothesis Hg(iii), and (3.98)). Since q<2, by choosing t∈(0,1) even smaller if necessary, we have
(3.101)ψ̂λ+(tũ)<0,
so
(3.102)ψ̂λ+(uλ)<0=ψ̂λ+(0)
(see (3.97)); hence uλ≠0.
From (3.97), we have
(3.103)(ψ̂λ+)′(uλ)=0,
so
(3.104)A(uλ)+(β+ĉ1)uλ=Nγλ+(uλ).
Acting on (3.104) with -uλ-∈H01(Ω), we obtain that uλ⩾0, uλ≠0. Also, acting on (3.104) with (uλ-uμ)+∈H01(Ω), we have
(3.105)〈A(uλ),(uλ-uμ)+〉+∫Ω(β+ĉ1)uλ(uλ-uμ)+dz=∫Ωγλ+(z,uλ)(uλ-uμ)+dz=∫Ω(λg(z,uμ)+f(z,uμ)+ĉ1uμ)(uλ-uμ)+dz⩽〈A(uμ),(uλ-uμ)+〉+∫Ω(β+ĉ1)uμ(uλ-uμ)+dz
(see (3.94) and (3.93)), so
(3.106)〈A(uλ-uμ),(uλ-uμ)+〉+∫Ω(β+ĉ1)(uλ-uμ)(uλ-uμ)+dz⩽0,
thus
(3.107)σ((uλ-uμ)+)+ĉ1‖(uλ-uμ)+‖22⩽0,
hence
(3.108)1ĉ0‖(uλ-uμ)+‖2⩽0
(see (2.14)) and so finally
(3.109)uλ⩽uμ.
So, we have proved that
(3.110)uλ∈[0,uμ]={u∈H01(Ω):0⩽u(z)⩽uμ(z)foralmostallz∈Ω}.
Hence (3.104) becomes
(3.111)A(uλ)+βuλ=λNg(uλ)+Nf(uλ)
(see (3.94)), so
(3.112)-Δuλ(z)+β(z)uλ(z)=λg(z,uλ(z))+f(z,uλ(z))inΩ,uλ|∂Ω=0,
thus uλ∈intC+ (as in the proof of Proposition 3.6) and it is a solution of ((P)λ).
Ifuλ≠u0, then this is the desired second nontrivial positive smooth solution of ((P)λ).
So, we may assume thatuλ=u0 and that there is no other solution of ((P)λ) in the order interval
(3.113)[u0,uλ]={u∈H01(Ω):u0(z)⩽u(z)⩽uμ(z)foralmostallz∈Ω}.
We introduce the following truncation of γλ+(z,·):
(3.114)γ̃λ+(z,ζ)={γλ+(z,u0(z))ifζ<u0(z),γλ+(z,ζ)ifu0(z)⩽ζ.
This is a Carathéodory function. We set
(3.115)Γ̃λ+(z,ζ)=∫0ζγ̃λ+(z,s)ds
and consider the C1-functional ψ̃λ+:H01(Ω)→ℝ, defined by
(3.116)ψ̃λ+(u)=12σ(u)+ĉ12‖u‖22-∫ΩΓ̃λ+(z,u(z))dz∀u∈H01(Ω).
From (3.94) and (3.114), it follows that ψ̃λ+ is coercive. Also, it is sequentially weakly lower semicontinuous. Hence, we can find ũλ∈H01(Ω), such that
(3.117)ψ̃λ+(ũλ)=inf⁡u∈H01(Ω)ψ̃λ+(u),
so
(3.118)(ψ̃λ+)′(ũλ)=0,
and thus
(3.119)A(ũλ)+(β+ĉ1)ũλ=Nγ̃λ+(ũλ).
As before, acting on (3.119) with (u0-ũλ)+∈H01(Ω) and with (ũλ-uμ)+∈H01(Ω), we show that ũλ∈[u0,uμ]. Then, from (3.94) and (3.114), it follows that
(3.120)A(ũλ)+βũλ=λNg(ũλ)+Nf(ũλ),
so ũλ∈intC+ is a solution of ((P)λ) in [u0,uμ], hence ũλ=u0.
Letϱ=∥u0∥∞ and let γϱ>0 and γ̂ϱ>0 be as postulated by hypotheses Hg(iv) and Hf(iv), respectively. We have
(3.121)-Δu0(z)+(β(z)+γ̂ϱ)u0(z)+λγϱu0(z)q-1=λg(z,u0(z))+λγϱu0(z)q-1+f(z,u0(z))+γ̂ϱu0(z)⩽λg(z,uμ(z))+λγϱuμ(z)q-1+f(z,uμ(z))+γ̂ϱuμ(z)⩽μg(z,uμ(z))+μγϱuμ(z)q-1+f(z,uμ(z))+γ̂ϱuμ(z)=-Δuμ(z)+(β(z)+γ̂ϱ)uμ(z)+μγϱuμ(z)q-1foralmostallz∈Ω
(see Hg(iv) and (iii), Hf(iv) and recall that u0⩽uμ and λ<μ), so there exists c12>0, such that
(3.122)Δ(uμ-u0)(z)⩽(‖β+‖∞+γ̂ϱ)(uμ-u0)(z)+μγϱc12(uμ-u0)(z),
for almost all z∈Ω (recall that the function ζ↦ζq-1 is locally Lipschitz); thus
(3.123)uμ-u0∈int⁡C+
(see Struwe [13] and Pucci and Serrin [14, page 120]). So, we have that
(3.124)uλ=u0∈int⁡C01(Ω¯)⁡[0,uμ].
Note that
(3.125)ψ̃λ+|[0,uμ]=ψλ+|[0,uμ]
(see (3.94) and (3.114)), so
(3.126)u0isalocalC01(Ω¯)-minimizerofψλ+
(see (3.124)), and thus
(3.127)u0isalocalH01(Ω)-minimizerofψλ+
(Brézis and Nirenberg [16]). This proves Claim 3.
By virtue of Claim3, as in Gasiński and Papageorgiou [17, proof of Theorem 3.4], we can find ϱλ∈(0,1) small, such that
(3.128)ψλ+(u0)<inf⁡⁡{ψλ+(u):‖u-u0‖=ϱλ}=ηλ+.
As in Proposition 3.4, for ũ∈intC+ with ∥ũ∥2=1, we have
(3.129)ψλ+(tũ)⟶-∞ast⟶+∞.
Note that φ̂λ+=ψλ+-ξλ+ with ξλ+∈ℝ. Hence by virtue of Proposition 3.1, ψλ+ satisfies the Cerami condition. This fact together with (3.128) and (3.129) permits the use of the mountain pass theorem (see Theorem 2.1). So, we can find û∈H01(Ω), such that
(3.130)ψλ+(u0)<ηλ+⩽ψλ+(û),(3.131)(ψλ+)′(û)=0.
From (3.130) we have û≠u0. From (3.131) and Claim 2, we have that
(3.132)u0⩽û.
Hence
(3.133)A(û)+βû=λNg(û)+Nf(û)
(see (3.84)), and so û∈intC+ solves problem ((P)λ).
Moreover, as before, using the strong maximum principle (see Vázquez [15] and Pucci and Serrin [14, page 120]), we have(3.134)û-u0∈int⁡C+.
In a similar way, usingv0∈-intC+, we define(3.135)hλ-(z,ζ)={λg(z,ζ)+f(z,ζ)+ĉ1ζifζ<v0(z),λg(z,v0(z))+f(z,v0(z))+ĉ1v0(z)ifv0(z)⩽ζ.
This is a Carathéodory function. We set
(3.136)Hλ-(z,ζ)=∫0ζhλ-(z,s)ds
and consider the C1-functional ψλ-:H01(Ω)→ℝ, defined by
(3.137)ψλ-(u)=12σ(u)+ĉ12‖u‖22-∫ΩHλ-(z,u(z))dz∀u∈H01(Ω).
Reasoning as above, using this time ψλ-, we obtain a second negative smooth solution v̂∈-intC+ of problem ((P)λ), such that
(3.138)v0-v̂∈int⁡C+.


## 4. Five Solutions

In this section, we prove two multiplicity theorems, establishing five nontrivial smooth solutions whenλ∈(0,λ*). In the second multiplicity theorem, we provide sign information for all the solutions (i.e., we show that the fifth solution is actually nodal).Theorem 4.1.
If hypothesesHg,Hf,H0, and Hβ hold and λ∈(0,λ*), then problem ((P)λ) has at least five nontrivial smooth solutions:
(4.1)u0,û∈int⁡C+,û-u0∈int⁡C+,v0,v̂∈-int⁡C+,v0-v̂∈int⁡C+y0∈C01(Ω¯),u0-y0∈int⁡C+,y0-v0∈int⁡C+.Proof.
From Proposition3.7, we already have four nontrivial smooth solutions of constant sign:
(4.2)u0,û∈int⁡C+,û-u0∈int⁡C+,v0,v̂∈-int⁡C+,v0-v̂∈int⁡C+.
We consider the following truncation perturbation of the reaction of problem ((P)λ):
(4.3)hλ*(z,ζ)={λg(z,v0(z))+f(z,v0(z))+ĉ1v0(z)ifζ<v0(z),λg(z,ζ)+f(z,ζ)ifv0(z)⩽ζ⩽u0(z),λg(z,u0(z))+f(z,u0(z))+ĉ1u0(z)ifu0(z)<ζ.
This is a Carathéodory function. We set
(4.4)Hλ*(z,ζ)=∫0ζhλ*(z,s)ds
and consider the C1-functional ψλ*:H01(Ω)→ℝ, defined by
(4.5)ψλ*(u)=12σ(u)+ĉ12‖u‖22-∫ΩHλ*(z,u(z))dz∀u∈H01(Ω).
From (4.3), it follows that ψλ* is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find y0∈H01(Ω), such that
(4.6)ψλ*(y0)=inf⁡u∈H01(Ω)ψλ*(u).
As in the proof of Proposition 3.7, using (3.98), we show that
(4.7)ψλ*(y0)<0=ψλ*(0),
hence y0≠0. From (4.6), we have
(4.8)(ψλ*)′(y0)=0,
so
(4.9)A(y0)+(β+ĉ1)y0=Nhλ*(y0).
On (4.9) we act with (y0-u0)+∈H01(Ω). Then
(4.10)〈A(y0),(y0-u0)+〉+∫Ω(β+ĉ1)y0(y0-u0)+dz=∫Ω(λg(z,u0)+f(z,u0)+ĉ1u0)(y0-u0)+dz=〈A(u0),(y0-u0)+〉+∫Ω(β+ĉ)u0(y0-u0)+dz
(see (4.3)), so
(4.11)〈A(y0-u0),(y0-u0)+〉+∫Ω(β+ĉ1)(y0-u0)(y0-u0)+dz=0,
thus
(4.12)σ((y0-u0)+)+ĉ1‖(y0-u0)+‖22=0,
and hence
(4.13)1ĉ0‖(y0-u0)+‖2⩽0
(see (2.14)); hence y0⩽u0.
Similarly, acting on (4.9) with (v0-y0)+∈H01(Ω), we show that v0⩽y0. Therefore,
(4.14)y0∈[v0,u0]={u∈H01(Ω):v0(z)⩽u(z)⩽u0(z)foralmostallz∈Ω},
and so (4.9) becomes
(4.15)A(y0)+βy0=λNg(y0)+Nf(y0)
(see (4.3)); thus
(4.16)y0∈C01(Ω¯)
(regularity theory; see Struwe [13]), and it solves problem ((P)λ).
Moreover, as in the proof of Proposition3.7, using hypotheses Hg(iv) and Hf(iv), we also show that
(4.17)u0-y0∈int⁡C+,y0-v0∈int⁡C+.Next we will improve the conclusion of Theorem4.1 and show that the fifth solution y0 is nodal (sign changing). To do this we need to strengthen a little bit the hypotheses on f(z,·).The new hypotheses off are the following:Hf′̲:
f:Ω×ℝ→ℝ is a Carathéodory function, such that f(z,0)=0 for almost all z∈Ω,(i), (iii), (iv) are the same as the corresponding hypothesesHf(i), (iii), (iv),(ii) we have(4.18)lim⁡ζ→±∞f(z,ζ)ζ=+∞uniformlyforalmostallz∈Ω.Remark 4.2.
HypothesisHf′ is a slight restricted version of hypothesis Hf. Note that if
(4.19)ξ̂(z,ζ)=f(z,ζ)ζ-2F(z,ζ)
and there exists a function ϑ̂*∈L1(Ω)+, such that
(4.20)ξ̂(z,ζ)⩽ξ̂(z,y)+ϑ̂*(z)foralmostallz∈Ω,all0⩽ζ⩽yory⩽ζ⩽0,
then (4.20) and Hf (ii) imply Hf′(ii) (see Li and Yang [10]). Also, note that hypotheses Hf(i), (ii), and (iii) imply that there exists c*>0, such that
(4.21)f(z,ζ)ζ+c*ζ2⩾0foralmostallz∈Ω,allζ∈R.We consider the following auxiliary Dirichlet problem:((Q)λ)-Δu(z)+β(z)u(z)=λc1|u(z)|q-2u(z)-c*u(z)inΩ,u|∂Ω=0.
Here c1>0 and q∈(1,2) are as in hypothesis Hg(iii) and c*>0 is as in (4.21).Proposition 4.3.
For everyλ>0, problem ((Q)λ) has a unique nontrivial positive solution u̲λ∈intC+, and by oddness, we have that -u̲λ=v¯λ∈-intC+ is the unique negative solution of ((Q)λ).Proof.
Letkλ+:Ω×ℝ→ℝ be the Carathéodory function, defined by
(4.22)kλ+(z,ζ)={0ifζ⩽0,λc1ζq-1-(ĉ1-c*)ζifζ>0.
Clearly, we can always assume that c*⩾ĉ1 (see (4.21)). Let
(4.23)Kλ+(z,ζ)=∫0ζkλ+(z,s)ds,
and consider the C1-functional θλ+:H01(Ω)→ℝ, defined by
(4.24)θλ+(u)=12σ(u)+ĉ12‖u‖22-∫ΩKλ+(z,u(z))dz∀u∈H01(Ω).
Using (2.14) and (4.22), we have
(4.25)θλ+(u)⩾12ĉ0‖u‖2+c*-ĉ12‖u+‖22-λc1q‖u+‖qq.
Since q<2, from (4.25), we infer that θλ+ is coercive. Also, it is sequentially weakly lower semicontinuous (recall that c*⩾ĉ1). So, by the Weierstrass theorem, we can find u̲λ∈H01(Ω), such that
(4.26)θλ+(u̲λ)=inf⁡u∈H01(Ω)θλ+(u).
As before (see the proof of Proposition 3.7), since q<2, we have
(4.27)θλ+(u̲λ)<0=θλ+(0),
hence u̲λ≠0. From (4.26), we have
(4.28)(θλ+)′(u̲λ)=0,
so
(4.29)A(u̲λ)+(β+ĉ1)u̲λ=Nkλ+(u̲λ).
Acting on (4.29) with -u̲λ-∈H01(Ω), we show that u̲λ⩾0,u̲λ≠0 (see (2.14)). Then (4.29) becomes
(4.30)A(u̲λ)+(β+c*)u̲λ=λc1u̲λq-1,
so
(4.31)-Δu̲λ(z)+(β(z)+c*)u̲λ(z)⩾0foralmostallz∈Ω,
thus
(4.32)Δu̲λ(z)⩽(‖β+‖∞+c*)u̲λ(z)foralmostallz∈Ω
(see hypothesis Hβ), and hence
(4.33)u̲λ∈int⁡C+
(see Vázquez [15] and Pucci and Serrin [14, page 120]).
We claim that this solution is the unique nontrivial positive solution of ((Q)λ). To this end, let u,v∈intC+ be two positive solutions of ((Q)λ). We have
(4.34)∫Ωλc1uq-1-c*uu(u2-v2)dz=∫Ω(λc1uq-1-c*u)(u-v2u)dz=∫Ω(-Δu)(u-v2u)dz+∫Ωβ(u2-v2)dz=∫Ω(∇u,∇u-∇(v2u))RNdz+∫Ωβ(u2-v2)dz=‖∇u‖22-∫Ω(∇u,2vu∇v-v2u2∇u)RNdz+∫Ωβ(u2-v2)dz=‖∇u‖22-∫Ω2vu(∇u,∇v)RNdz+∫Ωv2u2‖∇u‖2dz+∫Ωβ(u2-v2)dz.
Interchanging the roles of u and v in the above argument, we also have
(4.35)∫Ωλc1vq-1-c*vv(v2-u2)dz=‖∇v‖22-∫Ω2uv(∇v,∇u)RNdz+∫Ωu2v2‖∇v‖2dz+∫Ωβ(v2-u2)dz.
Adding (4.34) and (4.35), we obtain
(4.36)∫Ω(λc1uq-1-c*uu-λc1vq-1-c*vv)(u2-v2)dz=‖∇u-uv∇v‖22+‖∇v-vu∇u‖22⩾0.
Since q<2, the function ζ↦(λc1ζq-1-c*ζ)/ζ is strictly decreasing on (0,+∞). Hence, from (4.36), we infer that u=v. This proves the uniqueness of the nontrivial positive solution u̲λ∈intC+ of problem ((Q)λ).
The oddness of problem ((Q)λ) implies that v¯λ=-u̲λ∈-intC+ is the unique nontrivial negative solution of ((Q)λ).Using Proposition4.3, we can show that problem ((P)λ) (for λ∈(0,λ*)) has extremal constant sign solutions; that is, it has a smallest nontrivial positive solution and a biggest nontrivial negative solution.Proposition 4.4.
If hypothesesHg, Hf′, H0, and Hβ hold and λ∈(0,λ*), then problem ((P)λ) has a smallest nontrivial positive solution u+λ∈intC+ and a biggest nontrivial negative solution v-λ∈int-C+.Proof.
Letu¯ be a nontrivial positive solution of ((P)λ). From the proof of Proposition 3.6, we know that u¯∈intC+. We have
(4.37)-Δu¯(z)+β(z)u¯(z)=λg(z,u¯(z))+f(z,u¯(z))⩾λc1u¯(z)q-1-c*u¯(z),
for almost all z∈Ω (see hypothesis Hg(iii) and (4.21)).
We consider the following Carathéodory function:(4.38)k̂λ+(z,ζ)={0ifζ⩽0,λc1ζq-1+(ĉ1-c*)ζif0⩽ζ⩽u¯,λc1u¯(z)q-1+(ĉ1-c*)u¯(z)ifu¯<ζ.
Let
(4.39)K̂λ+(z,ζ)=∫0ζk̂λ+(z,s)ds,
and consider the C1-functional ξλ+:H01(Ω)→ℝ, defined by
(4.40)ξλ+(u)=12σ(u)+ĉ12‖u‖22-∫ΩK̂λ+(z,u(z))dz∀u∈H01(Ω).
From (4.38) and (2.14), it is clear that ξλ+ is coercive. Also ξλ+ is sequentially weakly lower semicontinuous. Thus we can find w0∈H01(Ω), such that
(4.41)ξλ+(w0)=inf⁡u∈H01(Ω)ξλ+(u).
As before, the presence of the “concave” term λc1ζq-1 implies that
(4.42)ξλ+(w0)<0=ξλ+(0),
that is, w0≠0. From (4.41), we have
(4.43)(ξλ+)′(w0)=0,
so
(4.44)A(w0)+(β+ĉ1)w0=Nk̂λ+(w0).
On (4.44) we act with -w0-∈H01(Ω) and obtain
(4.45)σ(w0-)+ĉ1‖w0-‖22=0
(see (4.38)), so
(4.46)1ĉ0‖w0-‖2⩽0
(see (2.14)), and hence w0⩾0, w0≠0.
Also, on (4.44) we act with (w0-u¯)+∈H01(Ω). Then(4.47)〈A(w0),(w0-u¯)+〉+∫Ω(β+ĉ1)w0(w0-u¯)+dz=∫Ωk̂λ+(z,w0)(w0-u¯)+dz=∫Ω(λc1u¯q-1-c*u¯)(w0-u¯)+dz⩽〈A(u¯),(w0-u¯)+〉+∫Ωβ((w0-u¯)+)dz+ĉ1‖(w0-u¯)+‖22
(see (4.38) and (4.37)), so
(4.48)σ((w0-u¯)+)+ĉ1‖(w0-u¯)+‖22⩽0,
thus
(4.49)1ĉ0‖(w0-u¯)+‖2⩽0
(see (2.14)), and hence
(4.50)w0⩽u¯.
So, we have proved that
(4.51)w0∈[0,u¯]={u∈H01(Ω):0⩽u(z)⩽u¯(z)foralmostallz∈Ω}.
This means that (4.44) becomes
(4.52)A(w0)+βw0=λc1w0q-1-c*w0,
so w0∈intC+ (regularity theory of Struwe [13] and strong maximum principle due to Vázquez [15] and Pucci Serrin [14, page 120]) and it solves problem ((Q)λ). Thus
(4.53)w0=u̲λ
(see Proposition 4.3), and
(4.54)u̲λ⩽u̲.
This shows that every nontrivial positive solution u of ((P)λ) satisfies
(4.55)u̲λ⩽u.
Similarly, we show that every nontrivial negative solution v of problem ((P)λ) satisfies
(4.56)v⩽v¯λ=-u̲λ.
LetS+(λ) (resp., S-(λ)) be the set of nontrivial positive (resp., negative) solutions of problem ((P)λ). Let C⊆S+(λ) be a chain (i.e., a totally ordered subset of S+(λ)). From Dunford and Schwartz [18, page 336], we can find a sequence {un}n⩾1⊆C, such that(4.57)inf⁡⁡C=inf⁡n⩾1un.
Lemma 1.5 of Heikkilä and Lakshmikantham [19, page 15] implies that we can have the sequence {un}n⩾1⊆C to be decreasing. Then we have
(4.58)A(un)+βun=λNg(un)+Nf(un),u̲λ⩽un⩽u1∀n⩾1,
so
(4.59)thesequnece{un}n⩾1⊆H01(Ω)isbounded.
Hence by passing to a suitable subsequence if necessary, we may assume that
(4.60)un⟶u*weaklyinH01(Ω),(4.61)un⟶u*inL2s′(Ω)andinLr(Ω).
So, passing to the limit as n→+∞ in (4.58) and using (4.61), we obtain
(4.62)A(u*)+βu*=λNg(u*)+Nf(u*),u̲λ⩽u*,
so
(4.63)inf⁡⁡C=u*∈S+(λ).
Since C was an arbitrary chain, invoking the Kuratowski-Zorn lemma, we infer that S+(λ) has a minimal element u+λ∈intC+. From Gasiński and Papageorgiou [17, Lemma 4.2, page 5763], we know that S+(λ) is downward directed (i.e., if u1,u2∈S+(λ); then we can find u∈S+(λ), such that u⩽u1 and u⩽u2). So, it follows that u+λ is the smallest nontrivial positive solution of problem ((P)λ).
Similarly, we introduce the biggest nontrivial negative solutionv-λ∈-intC+ of problem ((P)λ). Note that S-(λ) is upward directed (i.e., if v1,v2∈S-(λ), then we can find v∈S-(λ), such that v1⩽v and v2⩽v; see Gasiński and Papageorgiou [17, Lemma 4.3, page 5764]).Now that we have these extremal constant sign solutions, we can produce a nodal solution of problem ((P)λ) (with λ∈(0,λ*)).Theorem 4.5.
If hypothesesHg, Hf′, H0, and Hβ hold and λ∈(0,λ*), then problem ((P)λ) has at least five nontrivial smooth solutions:
(4.64)u0,û∈int⁡C+,û-u0∈int⁡C+,v0,v̂∈-int⁡C+,v0-v̂∈int⁡C+y0∈C01(Ω¯)nodalwithu0-y0∈int⁡C+,y0-v0∈int⁡C+.
Moreover, problem ((P)λ) has a smallest nontrivial positive solution and a biggest negative solution.Proof.
The existence of extremal nontrivial constant sing solutions is guaranteed by Proposition4.4. Let u+λ∈intC+ and v-λ∈-intC+ be these two extremal solutions.
We introduce the following truncation perturbation of the reaction of problem ((P)λ):
(4.65)γλ(z,ζ)={λg(z,v-λ(z))+f(z,v-λ(z))+ĉ1v-λ(z)ifζ<v-λ(z),λg(z,ζ)+f(z,ζ)+ĉ1ζifv-λ(z)⩽ζ⩽u+λ(z),λg(z,u+λ(z))+f(z,u+λ(z))+ĉ1u+λ(z)ifu+λ(z)<ζ.
This is a Carathéodory function. We set
(4.66)Γλ(z,ζ)=∫0ζγλ(z,s)ds
and consider the C1-functional χλ:H01(Ω)→ℝ, defined by
(4.67)χλ(u)=12σ(u)+ĉ12‖u‖22-∫ΩΓλ(z,u(z))dz∀u∈H01(Ω).
Also, we introduce
(4.68)γλ±(z,ζ)=γλ(z,±ζ±),Γλ±(z,ζ)=∫0ζγλ±(z,s)ds
and consider the C1-functionals χλ±:H01(Ω)→ℝ, defined by
(4.69)χλ±(u)=12σ(u)+ĉ12‖u‖22-∫ΩΓλ±(z,u(z))dz∀u∈H01(Ω).
As in the proof of Theorem 4.1, we show that
(4.70)Kχλ⊆[v-λ,u+λ],Kχλ+⊆[0,u+λ],Kχλ-⊆[v-λ,0]
(see (4.65)). The extremality of the solutions v-λ and u+λ implies that
(4.71)Kχλ⊆[v-λ,u+λ],Kχλ+={0,u+λ},Kχλ-={v-λ,0}.
Claim 4.
Solutionsu+λ and v-λ are both local minimizers of χλ.
Evidentlyχλ+ is coercive (see (4.65)) and sequentially weakly lower semicontinuous. So, we can find u+∈H01(Ω), such that
(4.72)χλ+(u+)=inf⁡u∈H01(Ω)χλ+(u).
As before (see the proof of Proposition 3.7), the presence of the “concave” term implies that
(4.73)χλ+(u+)<0=χλ+(0),
hence u+≠0, and so u+=u+λ (see (4.71)). Since
(4.74)χλ|C+=χλ+|C+,
it follows that u+=u+λ∈intC+ is local C01(Ω¯)-minimizers of χλ; hence by Brézis and Nirenberg [16], it is also local H01(Ω)-minimizers of χλ.
Similarly this is forv-λ using this time the functional χλ-. This proves the claim.
Without any loss of generality, we may assume that(4.75)χλ(v-λ)⩽χλ(u+λ).
The analysis is similar, if the opposite inequality holds. Because of the claim, we can find ϱ∈(0,1) small, such that
(4.76)χλ(v-λ)⩽χλ(u+λ)<inf⁡⁡{χλ(u):‖u-u+λ‖=ϱ}=ηλ
(see Gasiński and Papageorgiou [17, proof of Theorem 3.4]).
Sinceχλ is coercive (see (4.65)), it satisfies the Cerami condition. This fact and (4.76) permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find y0∈Kχλ⊆[v-λ,u+λ] (see (4.71)), such that
(4.77)ηλ⩽χλ(y0),
so
(4.78)y0∉{v-λ,u+λ}
(see (4.76)).
Sincey0 is a critical point of χλ of mountain pass type, we have
(4.79)C1(χλ,y0)≠0
(see e.g., Chang [20]). On the other hand, hypothesis Hf(iii) implies that we can find ξ̂1>0, such that
(4.80)f(z,ζ)ζ-μF(z,ζ)⩽ξ̂1ζ2foralmostallz∈Ω,all|ζ|⩽δ1,
for some δ1⩽δ0. This combined with hypothesis Hg(iii) implies that
(4.81)μλG(z,ζ)+μF(z,ζ)⩾λg(z,ζ)ζ+f(z,ζ)ζ>0foralmostallz∈Ω,all|ζ|⩽δ2,
for some δ2⩽δ1 and
(4.82)esssup⁡ΩλG(⋅,δ2)+F(⋅,δ2)>0.
Hence invoking Proposition 2.1 of Jiu and Su [21], we infer that
(4.83)Ck(χλ,0)=0∀k⩾0.
Combining (4.79) and (4.83), we have that y0≠0. Since y0∈[v-λ,u+λ], the extremality of v-λ and u+λ implies that y0 must be a nodal solution of problem ((P)λ), and the regularity theory (see Struwe [13]) implies that y0∈C01(Ω¯).



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