# Nonhomogeneous Nonlinear Dirichlet Problems with ap-Superlinear Reaction

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

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## Abstract

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reactionf(z,ζ), whose primitive f(z,ζ) is p-superlinear near ±∞, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the p-Laplacian, the (p,q)-Laplacian, and the p-generalized mean curvature operator.


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## Body

## 1. Introduction

The motivation for this paper comes from the work of Wang [1] on superlinear Dirichlet equations. More precisely, let Ω⊆ℝN be a domain with a C2-boundary ∂Ω. Wang [1] studied the following Dirichlet problem:(1.1)-Δu(z)=f(u(z))inΩ,u|∂Ω=0.
He assumes that f∈C1(ℝ), f(0)=f′(0)=0, |f′(ζ)|≤c(1+|ζ|r-2), with 1<r<2*, where(1.2)2*={2NN-2ifN≥3,+∞ifN=1,2
and that there exist μ>2 and M>0, such that(1.3)0<μF(ζ)≤f(ζ)ζ∀|ζ|≥M,
where(1.4)F(ζ)=∫0ζf(s)ds
(this is the so-called Ambrosetti-Rabinowitz condition). Under these hypotheses, Wang [1] proved that problem (1.1) has at least three nontrivial solutions.The aim of this work is to establish the result of Wang [1] for a larger class of nonlinear Dirichlet problems driven by a nonhomogeneous nonlinear differential operator. In fact, our formulation unifies the treatment of “superlinear’’ equations driven by the p-Laplacian, the (p,q)-Laplacian, and the p-generalized mean curvature operators. In addition, our reaction term f(z,ζ) is z dependent, need not be C1 in the ζ-variable, and in general does not satisfy the Ambrosetti-Rabinowitz condition. Instead, we employ a weaker “superlinear’’ condition, which incorporates in our framework functions with “slower’’ growth near ±∞. An earlier extension of the result of Wang [1] to equations driven by the p-Laplacian was obtained by Jiang [2, Theorem  12, p.1236] with a continuous “superlinear’’ reaction f(z,ζ) satisfying the Ambrosetti-Rabinowitz condition.So, letΩ⊆ℝN be as above. The problem under consideration is the following:(1.5)-div⁡a(∇u(z))=f(z,u(z))a.e.inΩ,u|∂Ω=0.
Here a:ℝN→ℝN is a map which is strictly monotone and satisfies certain other regularity conditions. The precise conditions on a are formulated in hypotheses H(a). These hypotheses are rather general, and as we already mentioned, they unify the treatment of various differential operators of interest. The reaction f(z,ζ) is a Carathéodory function (i.e., for all ζ∈ℝ, the function z↦f(z,ζ) is measurable and for almost all z∈Ω, the function ζ↦f(z,ζ) is continuous). We assume that the primitive(1.6)F(z,ζ)=∫0ζf(z,s)ds
exhibits p-superlinear growth near ±∞. However, we do not employ the usual in such cases, the Ambrosetti-Rabinowitz condition. Instead we use a weaker condition (see hypotheses H(f)), which permits the consideration of a broader class of reaction terms.Our approach is variational based on the critical point theory combined with Morse theory (critical groups). In the next section for easy reference, we present the main mathematical tools that we will use in the paper. We also state the precise hypotheses on the mapsa and f and explore some useful consequences of them.


## 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 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. Nevertheless, the Cerami condition suffices to have a deformation theorem, and from it the minimax theory of certain critical values ofφ is derive (see, e.g., Gasiński and Papageorgiou [3]). In particular, we can state the following theorem, known in the literature as the “mountain pass theorem.’’Theorem 2.1.
Ifφ∈C1(X) satisfies the Cerami condition, x0,x1∈X are such that ∥x1-x0∥>ϱ>0, and
(2.2)max⁡⁡{φ(x0),φ(x1)}<inf⁡{φ(x):‖x-x0‖=ϱ}=ηϱ,c=inf⁡γ∈Γmax⁡0≤t≤1⁡φ(γ(t)),
where
(2.3)Γ={γ∈C([0,1];X):γ(0)=x0,γ(1)=x1},
then c≥ηϱ and c is a critical value of φ.In the analysis of problem (1.5) in addition to the Sobolev space W01,p(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(·) denotes the outward unit normal on ∂Ω.In what follows, byλ̂1 we denote the first eigenvalue of (-Δp,W01,p(Ω)), where Δp denotes the p-Laplace operator, defined by
(2.7)Δpu=div⁡(‖∇u‖p-2∇u)∀u∈W01,p(Ω).
We know (see, e.g., Gasiński and Papageorgiou [3]) that λ̂1>0 is isolated and simple (i.e., the corresponding eigenspace is one-dimensional) and
(2.8)λ̂1=inf⁡{‖∇u‖pp‖u‖pp:u∈W01,p(Ω),u≠0}.
In this variational characterization of λ̂1, the infimum is realized on the corresponding one-dimensional eigenspace. From (2.8), we see that the elements of the eigenspace do not change sign. In what follows, by û1 we denote the Lp-normalized (i.e., ∥û1∥p=1) positive eigenfunction corresponding to λ̂1>0. The nonlinear regularity theory for the p-Laplacian equations (see, e.g., Gasiński and Papageorgiou [3, p. 737]) and the nonlinear maximum principle of Vázquez [4] imply that û1∈intC+.Now, letφ∈C1(X) and let c∈ℝ. We introduce the following notation:
(2.9)φc={x∈X:φ(x)≤c},Kφ={x∈X:φ′(x)=0},Kφc={x∈Kφ:φ(x)=c}.Let(Y1,Y2) be a topological pair with Y2⊆Y1⊆X. For every integer k≥0, by Hk(Y1,Y2) we denote the kth relative singular homology group for the pair (Y1,Y2) with integer coefficients. The critical groups of φ at an isolated point x0∈Kφ with φ(x0)=c (i.e., x0∈Kφc) are defined by
(2.10)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.Suppose thatφ∈C1(X) satisfies the Cerami condition and φ(Kφ)>-∞. Let c<infφ(Kφ). The critical groups of φ at infinity are defined by
(2.11)Ck(φ,∞)=Hk(X,φc)∀k≥0.
The second deformation theorem (see, e.g., Gasiński and Papageorgiou [3, p. 628]) guarantees that this definition is independent of the particular choice of the level c<infφ(Kφ).Suppose thatKφ is finite. We set
(2.12)M(t,x)=∑k≥0rank⁡Ck(φ,x)tk∀t∈R,x∈Kφ,P(t,∞)=∑k≥0rank⁡Ck(φ,∞)tk∀t∈R.
The Morse relation says that
(2.13)∑x∈KφM(t,x)=P(t,∞)+(1+t)Q(t),
where
(2.14)Q(t)=∑k≥0βktk
is a formal series in t∈ℝ with nonnegative integer coefficients βk∈ℕ.Now we will introduce the hypotheses on the mapsa(y) and f(z,ζ). So, let h∈C1(0,+∞) be such that
(2.15)0<th′(t)h(t)≤c0∀t>0,
for some c0>0 and
(2.16)c1tp-1≤h(t)≤c2(1+|t|p-1)∀t>0,
for some c1,c2>0.The hypotheses on the mapa(y) are the following:H(a)̲:a(y)=a0(∥y∥)y, where a0(t)>0 for all t>0 and(i)
a∈C(ℝN;ℝN)∩C1(ℝN∖{0};ℝN),(ii)
there existsc3>0, such that(2.17)‖∇a(y)‖≤c3h(‖y‖)‖y‖∀y∈RN∖{0},(iii)
we have(2.18)(∇a(y)ξ,ξ)RN≥h(‖y‖)‖y‖‖ξ‖2∀y∈RN∖{0},ξ∈RN,(iv)
ifG:ℝN→ℝN is a function, such that ∇G(y)=a(y) for y∈ℝN and G(0)=0, then there exists c4>0, such that(2.19)pG(y)-(a(y),y)RN≥-c4∀y∈RN.Remark 2.2.
Let(2.20)G0(t)=∫0ta0(s)sds∀t≥0.
Evidently G0 is strictly convex and strictly increasing on ℝ+=[0,+∞). We set
(2.21)G(y)=G0(‖y‖)∀y∈RN.
Then G is convex, G(0)=0, and
(2.22)∇G(y)=G0′(‖y‖)y‖y‖=a0(‖y‖)y=a(y)∀y∈RN∖{0}.
Hence the primitive function G(y) used in hypothesis H(a)(iv) is uniquely defined. Note that the convexity of G implies that
(2.23)G(y)≤(a(y),y)RN∀y∈RN.
HypothesesH(a) and (2.23) lead easily to the following lemma summarizing the main properties of a.Lemma 2.3.
If hypothesesH(a) hold, then(a)
the mapy↦a(y) is maximal monotone and strictly monotone,(b)
there existsc5>0, such that
(2.24)‖a(y)‖≤c5(1+‖y‖p-1)∀y∈RN,(c)
we have(2.25)(a(y),y)RN≥c1p-1‖y‖p∀y∈RN
(where c1>0 is as in (2.16)).From the above lemma and the integral form of the mean value theorem, we have the following result.Corollary 2.4.
If hypothesesH(a) hold, then there exists c6>0, such that
(2.26)c1p(p-1)‖y‖p≤G(y)≤c6(1+‖y‖p)∀y∈RN.Example 2.5.
The following maps satisfy hypothesesH(a):(a)
a(y)=∥y∥p-2y with 1<p<+∞.This map corresponds to the p-Laplace differential operator
(2.27)Δpu=div⁡(‖∇u‖p-2∇u)∀u∈W01,p(Ω).(b)
a(y)=∥y∥p-2y+μ∥y∥p-2y with 2≤q<p<+∞,μ≥0.This map corresponds to the (p,q)-Laplace differential operator
(2.28)Δpu+μΔqu∀u∈W01,p(Ω).
This is an important operator occurring in quantum physics (see Benci et al. [5]). Recently there have been some papers dealing with the existence and multiplicity of solutions for equations driven by such operators. We mention the works of Cingolani and Degiovanni [6], Figueiredo [7], and Sun [8]:(c)
a(y)=(1+∥y∥2)(p-2)/2y, with 2≤p<+∞.This map corresponds to the p-generalized mean curvature operator
(2.29)div⁡((1+‖∇u‖2)(p-2)/2∇u)∀u∈W01,p(Ω).
Such equations were investigated by Chen-Shen [9]:(d)
a(y)=∥y∥p-2y+∥y∥p-2y/(1+∥y∥p), with 1<p<+∞,(e)
a(y)=∥y∥p-2y+ln(1+∥y∥p-2)y, with 2≤p<+∞.
LetA:W01,p(Ω)→W-1,p′(Ω)=W01,p(Ω)* (with 1/p+1/p′=1) be the nonlinear map, defined by
(2.30)〈A(u),y〉=∫Ω(a(∇u(z)),∇y(z))RNdz∀u,y∈W01,p(Ω).
From Gasiński and Papageorgiou [10, Proposition  3.1, p. 852], we have the following result for this map.Proposition 2.6.
If hypothesesH(a) hold, then the map A:W01,p(Ω)→W-1,p′(Ω) defined by (2.30) is bounded, continuous, strictly monotone, hence maximal monotone too, and of type (S)+; that is, if un→u weakly in W01,p(Ω) and
(2.31)limsup⁡n→+∞⁡〈A(un),un-u〉≤0,
then un→u in W01,p(Ω).The next lemma is an easy consequence of (2.8) and of the fact that û1∈intC+ (see, e.g., Papageorgiou and Kyritsi-Yiallourou [11, p. 356]).Proposition 2.7.
Ifϑ∈L∞(Ω)+, ϑ(z)≤(c1/p(p-1))λ̂1 for almost all z∈Ω, ϑ≠(c1/p(p-1))λ̂1, then there exists ξ0>0, such that
(2.32)c1p-1‖∇u‖pp-∫Ωϑ(z)|u(z)|pdz≥ξ0‖∇u‖pp∀u∈W01,p(Ω).
The hypotheses on the reactionf(z,ζ) are the following:
H(f)̲:f:Ω×ℝ→ℝ is a Carathéodory function, such that f(z,0)=0 for almost all z∈Ω and(i)
there exista∈L∞(Ω)+, c>0 and r∈(p,p*), with
(2.33)p*={NpN-pifp<N,+∞ifp≥N,
such that
(2.34)|f(z,ζ)|≤a(z)+c|ζ|r-1for almost allz∈Ω,allζ∈R;(ii)
if(2.35)F(z,ζ)=∫0ζf(z,s)ds,
then
(2.36)lim⁡ζ→±∞⁡F(z,ζ)|ζ|p=+∞uniformly for almost allz∈Ω,(iii)
there existτ∈((r-p)max{1,N/p},p*) and β0>0, such that
(2.37)lim⁡inf⁡ζ→±∞⁡f(z,ζ)ζ-pF(z,ζ)|ζ|τ≥β0uniformly for almost allz∈Ω,(iv)
there existsϑ∈L∞(Ω)+, such that ϑ(z)≤(c1/p(p-1))λ̂1 for almost all z∈Ω, ϑ≠(c1/p(p-1))λ̂1, and
(2.38)limsup⁡ζ→0⁡pF(z,ζ)|ζ|p≤ϑ(z)uniformly for almost allz∈Ω;(v)
for everyϱ>0, there exists ξϱ>0, such that
(2.39)f(z,ζ)ζ+ξϱ|ζ|p≥0for almost allz∈Ω,all|ζ|≤ϱ.Remark 2.8.
HypothesisH(f)(ii) implies that for almost all z∈Ω, the primitive F(z,·) is p-superlinear. Evidently, this condition is satisfied if
(2.40)lim⁡ζ→±∞f(z,ζ)|ζ|p-2ζ=+∞uniformly for almost allz∈Ω,
that is, for almost all z∈Ω, the reaction f(z,·) is (p-1)-superlinear. However, we do not use the usual for “superlinear’’ problems, the Ambrosetti-Rabinowitz condition. We recall that this condition says that there exist μ>0 and M>0, such that
(2.41)0<μF(z,ζ)≤f(z,ζ)ζfor almost allz∈Ω,all|ζ|≥M,ess supΩF(⋅,M)>0.
Integrating (2.41), we obtain the weaker condition
(2.42)c7|ζ|μ≤F(z,ζ)for almost allz∈Ω,all|ζ|≥M,
with c7>0. Evidently from (2.42) we have the much weaker condition
(2.43)lim⁡ζ→±∞F(z,ζ)|ζ|p=+∞uniformly for almost allz∈Ω.
Here, the p-superlinearity condition (2.43) is coupled with hypothesis H(f)(iii), which is weaker than the Ambrosetti-Rabinowitz condition (2.41). Indeed, suppose that the Ambrosetti-Rabinowitz condition is satisfied. We may assume that μ>(r-p)max{1,N/p}. Then
(2.44)f(z,ζ)ζ-pF(z,ζ)|ζ|μ=f(z,ζ)ζ-μF(z,ζ)|ζ|μ+(μ-p)F(z,ζ)|ζ|μ≥(μ-p)c7
(see (2.41) and (2.42)). Therefore, hypothesis H(f)(iii) is satisfied.Example 2.9.
The following function satisfies hypothesesH(f), but not the Ambrosetti-Rabinowitz condition. For the sake of simplicity we drop the z-dependence:
(2.45)f(ζ)={|ζ|q-2ζif|ζ|≤1,p|ζ|p-2ζ(ln⁡|ζ|+1p)if|ζ|>1,
with 1<p<q<+∞.
In this work, for everyu∈W01,p(Ω), we set
(2.46)‖u‖=‖∇u‖p
(by virtue of the Poincaré inequality). We mention that the notation ∥·∥ will also be used to denote the ℝN-norm. However, no confusion is possible, since it is always clear from the context, whose norm is used. For every ζ∈ℝ, we set
(2.47)ζ±=max⁡⁡{±ζ,0},
and for u∈W01,p(Ω), we define
(2.48)u±(⋅)=u(⋅)±.
Then u±∈W01,p(Ω), and we have
(2.49)u=u+-u-,|u|=u++u-.
By |·|N we denote the Lebesgue measure on ℝN. Finally, for a given measurable function h:Ω×ℝ→ℝ (e.g., a Carathéodory function), we define
(2.50)Nh(u)(⋅)=h(⋅,u(⋅))∀u∈W01,p(Ω)
(the Nemytskii map corresponding to h(·,·)).


## 3. Three-Solution Theorem

In this section, we prove a multiplicity theorem for problem (1.5), producing three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative).First we produce two constant sign solutions of (1.5). For this purpose, we introduce the positive and negative truncations of f(z,·), namely:(3.1)f±(z,ζ)=f(z,±ζ±)∀(z,ζ)∈Ω×R.
Both are Carathéodory functions. We set(3.2)F±(z,ζ)=∫0ζf±(z,s)ds
and consider the C1-functionals φ±:W01,p(Ω)→ℝ, defined by(3.3)φ±(u)=∫ΩG(∇u(z))dz-∫ΩF±(z,u(z))dz∀u∈W01,p(Ω).
Also, let φ:W01,p(Ω)→ℝ be the C1-energy functional for problem (1.5), defined by(3.4)φ(u)=∫ΩG(∇u(z))dz-∫ΩF(z,u(z))dz∀u∈W01,p(Ω).Proposition 3.1.
If hypothesesH(a) and H(f) hold, then the functionals φ± satisfy the Cerami condition.Proof.
We do the proof forφ+, the proof for φ- being similar.
So, let{un}n≥1⊆W01,p(Ω) be a sequence, such that
(3.5)|φ+(un)|≤M1∀n≥1,
for some M1>0, and
(3.6)(1+‖un‖)φ′+(un)⟶0inW-1,p′(Ω).
From (3.6), we have
(3.7)|〈φ′+(un),v〉|≤ɛn‖v‖1+‖un‖∀v∈W01,p(Ω),
with ɛn↘0, so
(3.8)|〈A(un),v〉-∫Ωf+(z,un)vdz|≤ɛn‖v‖1+‖un‖∀n≥1.
In (3.8), first we choose v=-un-∈W01,p(Ω). Then using Lemma 2.3(c), we have
(3.9)c1p-1‖∇un-‖pp≤ɛn∀n≥1,
so
(3.10)un-⟶0inW01,p(Ω).
Next, in (3.8) we choose v=un+∈W01,p(Ω). We obtain
(3.11)-∫Ω(a(∇un+),∇un+)RN+∫Ωf(z,un+)un+dz≤ɛn∀n≥1.
From (3.5) and (3.10), we have
(3.12)∫ΩpG(∇un+)dz-∫ΩpF(z,un+)dz≤M2∀n≥1
for some M2>0. Adding (3.11) and (3.12), we obtain
(3.13)∫Ω(pG(∇un+)-(a(∇un+),∇un+)RN)dz+∫Ω(f(z,un+)un+-pF(z,un+))dz≤M3∀n≥1,
for some M3>0, so
(3.14)∫Ω(f(z,un+)un+-pF(z,un+))dz≤M4=M3+c4|Ω|N∀n≥1
(see hypothesis H(a)(iv)).
HypothesesH(f)(i) and (iii) imply that we can find β1∈(0,β0) and c8>0, such that
(3.15)β1|ζ|τ-c8≤f(z,ζ)ζ-pF(z,ζ)for almost allz∈Ω,allζ∈R.
Using (3.15) and (3.14), we obtain
(3.16)β1‖un+‖ττ≤M5∀n≥1,
with M5=M4+c8|Ω|N>0 and so
(3.17)the sequence{un+}n≥1⊆Lτ(Ω)is bounded.
First suppose that N≠p. From hypothesis H(f)(iii), it is clear that we can always assume that τ≤r<p*. So, we can find t∈[0,1), such that
(3.18)1r=1-tτ+tp*.
Invoking the interpolation inequality (see, e.g., Gasiński and Papageorgiou [3, p. 905]), we have
(3.19)‖un+‖r≤‖un+‖τ1-t‖un+‖p*t,
so
(3.20)‖un+‖rr≤M6‖un+‖tr∀n≥1,
for some M6>0 (see (3.17) and use the Sobolev embedding theorem).
Recall that(3.21)|∫Ω(a(∇un+),∇un+)RNdz-∫Ωf(z,un+)un+dz|≤ɛn∀n≥1.
From hypothesis H(f)(i), we have
(3.22)f(z,ζ)ζ≤â(z)+ĉ|ζ|rfor almost allz∈Ω,allζ∈R,
with â∈L∞(Ω)+, ĉ>0. Therefore, from (3.21) and Lemma 2.3(c), we have
(3.23)c1p-1‖∇un+‖pp≤c9(1+‖un+‖rr)∀n≥1,
for some c9>0 and so
(3.24)‖un+‖p≤c10(1+‖un+‖tr)∀n≥1,
for some c10>0 (see (3.20)). The hypothesis on τ (see H(f)(iii)) implies that tr<p, and so
(3.25)the sequence{un+}n≥1⊆W01,p(Ω)is bounded,
and thus
(3.26)the sequence{un}n≥1⊆W01,p(Ω)is bounded
(see (3.26)).
Now, suppose thatN=p. In this case, we have p*=+∞, while from the Sobolev embedding theorem, we have that W01,p(Ω)⊆Lq(Ω) for all q∈[1,+∞). So, we need to modify the previous argument. Let ϑ>r≥τ. Then we choose t∈[0,1), such that
(3.27)1r=1-tτ+tϑ,
so
(3.28)tr=ϑ(r-τ)ϑ-τ.
Note that
(3.29)ϑ(r-τ)ϑ-τ⟶r-τasϑ⟶+∞=p*.
Since N=p, we have r-τ<p (see H(f)(iii)). Therefore, for large ϑ>r, we have that tr<p (see (3.28)). Hence, if in the previous argument, we replace p* with such a large ϑ>r, again we reach (3.26).
Because of (3.26), we may assume that
(3.30)un⟶uweakly inW01,p(Ω),(3.31)un⟶uinLp(Ω).
In (3.8), we choose v=un-u∈W01,p(Ω), pass to the limit as n→+∞, and use (3.30). Then
(3.32)lim⁡inf⁡n→+∞⁡〈A(un),un-u〉=0,
so
(3.33)un⟶uinW01,p(Ω)
(see Proposition 2.6). This proves that φ+ satisfies the Cerami condition.
Similarly we show thatφ- satisfies the Cerami condition.With some obvious minor modifications in the above proof, we can also have the following result.Proposition 3.2.
If hypothesesH(a) and H(f) hold, then the functional φ satisfies the Cerami condition.Next we determine the structure of the trivial critical pointu=0 for the functionals φ± and φ.Proposition 3.3.
If hypothesesH(a) and H(f) hold, then u=0 is a local minimizer for the functionals φ± and φ.Proof.
By virtue of hypothesesH(f)(i) and (iv), for a given ɛ>0 we can find cɛ>0, such that
(3.34)F(z,ζ)≤1p(ϑ(z)+ɛ)|ζ|p+cɛ|ζ|rfor almost allz∈Ω,allζ∈R.
Then for all u∈W01,p(Ω), we have
(3.35)φ+(u)=∫ΩG(∇u)dz-∫ΩF(z,u+)dz≥c1p(p-1)‖∇u‖pp-1p∫Ωϑ(u+)pdz-ɛpλ̂1‖u‖p-c11‖u‖r≥1p(ξ0-ɛλ̂1)‖u‖p-c11‖u‖r,
for some c11>0 (see Corollary 2.4, (2.8), (3.34), and Proposition 2.7). Choosing ɛ∈(0,λ̂1ξ0), we have
(3.36)φ+(u)≥c12‖u‖p-c11‖u‖r∀u∈W01,p(Ω),
for some c12>0. Since r>p, from (3.36), it follows that we can find small ϱ∈(0,1), such that
(3.37)φ+(u)>0∀u,with0<‖u‖≤ϱ,
so
(3.38)u=0is a local minimizer ofφ+.
Similarly, we show that u=0 is a local minimizer for the functionals φ- and φ.We may assume thatu=0 is an isolated critical point of φ+ (resp., φ-). Otherwise, we already have a sequence of distinct positive (resp., negative) solutions of (1.5) and so we are done. Moreover, as in Gasiński and Papageorgiou [12, proof of Theorem  3.4,] we can find small ϱ±∈(0,1), such that(3.39)inf⁡{φ±(u):‖u‖=ϱ±}=η±>0.
By virtue of hypothesis H(f)(ii) (the p-superlinear condition), we have the next result, which completes the mountain pass geometry for problem (1.1).Proposition 3.4.
If hypothesesH(a) and H(f) hold and u∈intC+, then φ±(tu)→-∞ as t→±∞.Proof.
By virtue of hypothesesH(f)(i) and (ii), for a given ξ>0, we can find c13=c13(ξ)>0, such that
(3.40)ξ|ζ|p-c13≤F(z,ζ)for almost allz∈Ω,allζ∈R.
Then for u∈intC+ and t>0, we have
(3.41)φ+(tu)=∫ΩG(t∇u)dz-∫ΩF(z,tu)dz≤c14(1+tp‖u‖p)-ξtp‖u‖pp+c13|Ω|N=tp(c14‖u‖p-ξ‖u‖pp)+c15,
for some c14>0 and with c15=c14+c13|Ω|N>0 (see Corollary 2.4 and (3.40)).
Choosingξ>c14(∥u∥p/∥u∥pp), from (3.41), it follows that
(3.42)φ+(tu)⟶-∞ast⟶+∞.
Similarly, we show that
(3.43)φ-(tu)⟶-∞ast⟶-∞.Now we are ready to produce two constant sign smooth solutions of (1.5).Proposition 3.5.
If hypothesesH(a) and H(f) hold, then problem (1.5) has at least two nontrivial constant sign smooth solutions
(3.44)u0∈int⁡C+,v0∈-int⁡C+.Proof.
From (3.39), we have
(3.45)φ+(0)=0<inf⁡{φ+(u):‖u‖=ϱ+}=η+.
Moreover, according to Proposition 3.4, for u∈intC+, we can find large t>0, such that
(3.46)φ+(tu)≤φ+(0)=0<η+,‖tu‖>ϱ+.
Then because of (3.45), (3.46), and Proposition 3.1, we can apply the mountain pass theorem (see Theorem 2.1) and find u0∈W01,p(Ω), such that
(3.47)φ+(0)=0<η+≤φ+(u0),(3.48)φ′+(u0)=0.
From (3.47) we see that u0≠0. From (3.48), we have
(3.49)A(u0)=Nf+(u0).
On (3.49) we act with -u0-∈W01,p(Ω) and obtain
(3.50)c1p-1‖∇u0-‖pp≤0
(see Lemma 2.3(c)), so
(3.51)u0≥0,u0≠0.
Then, from (3.49), we have
(3.52)-div⁡a(∇u0(z))=f(z,u0(z))a.e.inΩ,u0|∂Ω=0.
Theorem  7.1 of Ladyzhenskaya and Uraltseva [13, p. 286] implies that u0∈L∞(Ω). Then from Lieberman [14, p. 320], we have that u0∈C01,α(Ω¯) for some α∈(0,1). Let ϱ=∥u0∥∞, and let ξϱ>0 be as postulated by hypothesis H(f)(v). Then
(3.53)-div⁡a(∇u0(z))+ξϱu0(z)p-1≥0for almost allz∈Ω
(see (3.52) and hypothesis H(f)(iv)), so
(3.54)div⁡a(∇u0(z))≤ξϱu0(z)p-1for almost allz∈Ω.
Then, from Theorem  5.5.1 of Pucci and Serrin [15, p. 120], we have that u0∈intC+.
Similarly, working withφ-, we obtain another constant sign smooth solution v0∈-intC+.Next, using the Morse theory (critical groups), we will produce a third nontrivial smooth solution. To this end, first we compute the critical groups ofφ± at infinity (see also Wang [1] and Jiang [2]).Proposition 3.6.
If hypothesesH(a) and H(f) hold, then
(3.55)Ck(φ±,∞)=0∀k≥0.Proof.
We do the proof forφ+, the proof for φ- being similar.
By virtue of hypothesesH(f)(i) and (ii), for a given ξ>0, we can find c16=c16(ξ)>0, such that
(3.56)F+(z,ζ)≥ξ(ζ+)p-c16for almost allz∈Ω,allζ∈R.
Let
(3.57)E+={u∈∂B1:u+≠0},
where
(3.58)∂B1={u∈W01,p(Ω):‖u‖=1}.
For u∈E+ and t>0, we have
(3.59)φ+(tu)=∫ΩG(t∇u)dz-∫ΩF+(z,tu)dz≤c17(1+tp)-ξtp‖u+‖pp-c16|Ω|N=tp(c17-ξ‖u+‖pp)+c17+c16|Ω|N
for some c17>0 (see Corollary 2.4, (3.56) and recall that ∥u∥=∥∇u∥p=1).
Choosingξ>c17/∥u+∥pp, we see that
(3.60)φ+(tu)⟶-∞ast⟶+∞.
Hypothesis H(f)(iii) implies that we can find β1∈(0,β0) and M7>0, such that
(3.61)f+(z,ζ)ζ-pF(z,ζ)≥β1ζτfor almost allz∈Ω,allζ≥M7.
Then for all y∈W01,p(Ω), we have
(3.62)∫Ω(pF+(z,y)-f+(z,y)y)dz≤-∫{y≥M7}β1yτdz+c18,
for some c18>0 (see (3.61)). Let c19=c18+c4|Ω|N>0 (see hypothesis H(a)(iv)) and choose γ<-c19. Because of (3.60), for u∈E+ and for large t>0, we have
(3.63)φ+(tu)=∫ΩG(t∇u)dz-∫ΩF+(z,tu)dz≤γ.
Since φ+(0)=0, we can find t*>0, such that
(3.64)φ+(t*u)=γ.
Also, for large t>0, we have
(3.65)ddtφ+(tu)=〈φ′+(tu),u〉=1t〈φ′+(tu),tu〉=1t(∫Ω(a(t∇u),t∇u)RNdz-∫Ωf+(z,tu)tudz)≤1t(∫ΩpG(t∇u)dz+c4|Ω|N-∫ΩpF+(z,tu)dz+c18)≤1t(γ+c18+c4|Ω|n)<0
(see hypothesis H(a)(iv), (3.62), (3.63), and recall that pφ+(tu)≤φ+(tu)≤γ<0). Hence, by the implicit function theorem, t* is unique and in fact there is a unique function μ+∈C(E+), such that
(3.66)φ+(μ+(u)u)=γ∀u∈E+.
Let
(3.67)D+={u∈W01,p(Ω):u+≠0}.
We set
(3.68)μ̂+(u)=1‖u‖μ+(u‖u‖)∀u∈D+.
Then μ̂+∈C(D+) and
(3.69)φ+(μ̂+(u)u)=γ∀u∈D+.
Moreover, if φ+(u)=γ, then μ̂+(u)=1. We set
(3.70)μ̃+(u)={1ifφ+(u)≤γ,μ̂+(u)ifφ+(u)>γ.
Evidently μ̂+∈C(E+). Let h+:[0,1]×D+→D+ be defined by
(3.71)h+(t,u)=(1-t)u+tμ̃+(u)u.
Clearly h+ is continuous and
(3.72)h+(0,u)=u,h+(1,u)=μ̃+(u)u∈φ+γ,h+(t,u)=u∀t∈[0,1],u∈φ+γ
(see (3.70)) and so
(3.73)φ+γis a strict deformation retract ofD+.
It is easy to see that D+ is contractible in itself. Hence
(3.74)Hk(W01,p(Ω),D+)=0∀k≥0
(see Granas and Dugundji [16, p. 389]), so
(3.75)Hk(W01,p(Ω),φ+γ)=0∀k≥0
(see (3.73)) and thus
(3.76)Ck(φ+,∞)=0∀k≥0
(choosing γ<0 negative enough).
The same applies forφ-, using this time the sets
(3.77)E-={u∈∂B1:u-≠0},D-={u∈W01,p(Ω):u-≠0}.With suitable changes in the above proof, we can have the following result.Proposition 3.7.
If hypothesesH(a) and H(f) hold, then
(3.78)Ck(φ,∞)=0∀k≥0.Proof.
As before, hypothesesH(f)(i) and (ii) imply that for a given ξ>0, we can find c20=c20(ξ)>0, such that
(3.79)F(z,ζ)≥ξ|ζ|p-c20for almost allz∈Ω,allζ∈R.
Let u∈∂B1={u∈W01,p(Ω):∥u∥=1} and t>0. Then
(3.80)φ(tu)=∫ΩG(t∇u)dz-∫ΩF(z,tu)dz≤c21(1+tp)-ξtp‖u‖pp+c20|Ω|N≤tp(c21-ξ‖u‖pp)+c21+c20|Ω|N
for some c21>0 (see Corollary 2.4, (3.79) and recall that ∥u∥=1). Choosing ξ>c21/∥u∥pp, we see that
(3.81)φ(tu)⟶-∞ast⟶+∞.
Hypothesis H(f)(iii) implies that we can find β1∈(0,β0) and M8>0, such that
(3.82)f(z,ζ)ζ-pF(z,ζ)≥β1|ζ|τfor almost allz∈Ω,all|ζ|≥M8.
Then, for any y∈W01,p(Ω), we have
(3.83)∫Ω(pF(z,y)-f(z,y)y)d=∫{|y|<M8}(pF(z,y)-f(z,y)y)dz+∫{|y|≥M8}(pF(z,y)-f(z,y)y)dz≤-∫{|y|≥M8}β|y|τdz+c22,
for some c22>0 (see hypothesis H(f)(i)).
Letc23=c22+c4|Ω|N (see hypothesis H(a)(iv)) and choose γ<-c23. Because of (3.81), for a given u∈∂B1 and for large t>0, we have
(3.84)φ(tu)=∫ΩG(t∇u)dz-∫ΩF(z,tu)dz≤γ.
We also have
(3.85)ddtφ(tu)=〈φ′(tu),u〉=1t〈φ′(tu),tu〉=1t(∫Ω(a(t∇u),t∇u)RN-∫Ωf(z,tu)tudz)≤1t(∫ΩpG(t∇u)dz+c4|Ω|N-∫ΩpF(z,tu)dz+c22)≤1t(γ+c22+c4|Ω|N)<0
(see hypothesis H(f)(iv), (3.83), (3.84) and note that pφ(tu)≤φ(tu)≤γ<0).
The implicit function theorem implies that there exists uniqueμ∈C(∂B1), such that
(3.86)φ(μ(u)u)=γ∀u∈∂B1.
We define
(3.87)μ̂(u)=1‖u‖μ(u‖u‖)∀u≠0.
Then μ̂∈C(W01,p(Ω)∖{0}) and
(3.88)φ(μ̂(u)u)=γ∀u∈W01,p(Ω)∖{0}.
Moreover, if φ(u)=γ, then μ̂(u)=1. We introduce
(3.89)μ̃(u)={1ifφ(u)≤γ,μ̂(u)ifφ(u)>γ.
Evidently ũ∈C(W01,p(Ω)∖{0}). Let h:[0,1]×(W01,p(Ω)∖{0})→W01,p(Ω)∖{0} be defined by
(3.90)h(t,u)=(1-t)u+tμ̃(u)u.
Then
(3.91)h(0,u)=u,h(1,u)=μ̃(u)u∈φγ,h(t,u)=u∀t∈[0,1],u∈φγ,
so
(3.92)φγis a strong deformation retract ofW01,p(Ω)∖{0}.
Via the radial retraction, we see that ∂B1 is a retract of W01,p(Ω)∖{0} and W01,p(Ω)∖{0} is deformable into ∂B1. Invoking Theorem  6.5 of Dugundji [17, p. 325], we have that
(3.93)∂B1is a deformation retract ofW01,p(Ω)∖{0},
so
(3.94)∂B1andφγare homotopy equivalent.
Thus
(3.95)Hk(W01,p(Ω),∂B1)=Hk(W01,p(Ω),φγ)∀k≥0,
and finally
(3.96)Ck(φ,∞)=Hk(W01,p(Ω),∂B1)∀k≥0.
But ∂B1 is an absolute retract of W01,p(Ω) (see, e.g., Gasiński and Papageorgiou [3, p. 691]), hence contractible in itself. Therefore
(3.97)Hk(W01,p(Ω),∂B1)=0∀k≥0,
so
(3.98)Ck(φ,∞)=0∀k≥0.Now we are ready to produce the third nontrivial smooth solution for problem (1.5).Theorem 3.8.
If hypothesesH(a) and H(f) hold, then problem (1.5) has at least three nontrivial smooth solutions
(3.99)u0∈int⁡C+,v0∈-int⁡C+,y0∈C01(Ω¯)∖{0}.Proof.
From Proposition3.5, we already have two nontrivial constant sign and smooth solutions
(3.100)u0∈int⁡C+,v0∈-int⁡C+.
We assume that
(3.101)Kφ={0,u0,v0}.
Otherwise we already have a third nontrivial solution y0, and by the nonlinear regularity theory, y0∈C01(Ω¯), so we completed the proof.
Claim 1.
Ck(φ+,u0)=Ck(φ-,v0)=δk,1ℤ for all k≥0.
We do the proof for the pair(φ+,u0), the proof for the pair (φ-,v0) being similar.
We start by noting thatKφ+={0,u0}. Indeed, suppose that u∈Kφ+∖{0}. Then
(3.102)A(u)=Nf+(u).
Acting with -u-∈W01,p(Ω), we obtain
(3.103)c1p-1‖∇u-‖pp≤0
(see Lemma 2.3(c)) and so u0≥0, u0≠0. Moreover, by nonlinear regularity (see Ladyzhenskaya and Uraltseva [13] and Lieberman [14]), we have that u∈C+∖{0}. Since φ′|C+=φ′+|C+, we infer that u∈Kφ={0,u0,v0}, and hence u=u0.
Chooseγ,ϑ∈ℝ, such that
(3.104)γ<0=φ+(0)<ϑ<η+≤φ+(u0)
(see (3.47)), and consider the following set:
(3.105)φ+γ⊆φ+ϑ⊆W01,p(Ω)=W.
We consider the long exact sequence of singular homology groups corresponding to the above triple. We have
(3.106)⋯⟶Hk(W,φ+γ)→i*Hk(W,φ+ϑ)→∂*Hk-1(φ+ϑ,φ+γ)⟶⋯∀k≥0,
where i* is the group homomorphism induced by the embedding i:φ+γ→φ+ϑ and ∂* is the boundary homomorphism. Note that
(3.107)Hk(W,φ+γ)=Ck(φ+,∞)∀k≥0
(since γ<0, Kφ+={0,u0} and 0=φ+(0)<η+≤φ+(u0); see (3.47)), so
(3.108)Hk(W,φ+γ)=0∀k≥0
(see Proposition 3.6).
From the choice ofϑ>0, the only critical value of φ+ in the interval (γ,ϑ) is 0. Hence
(3.109)Hk-1(φ+ϑ,φ+γ)=Ck-1(φ+,0)=δk-1,0Z=δk,1Z∀k≥0
(see Proposition 3.3).
Finally, for the same reason, we have(3.110)Hk(W,φ+ϑ)=Ck(φ+,u0)∀k≥0.
From (3.108), (3.109), and (3.110), it follows that in (3.106) only the tail (i.e., k=1) is nontrivial. The rank theorem implies that
(3.111)rank⁡H1(W,φ+ϑ)=rank im∂*+rank ker∂*=rank im∂*+rank imi*=1+0=1
(by virtue of the exactness of (3.106)), so
(3.112)rank⁡C1(φ+,u0)≤1
(see (3.110)).
Butu0 is a critical point of mountain pass type for φ+. Hence
(3.113)rank⁡C1(φ+,u0)≥1.
From (3.112) and (3.113) and since
(3.114)Ck(φ+,u0)=Hk(W,φ+ϑ)=0∀k≠1,
we infer that
(3.115)Ck(φ+,u0)=δk,1Z∀k≥0.
Similarly, we show that(3.116)Ck(φ-,v0)=δk,1Z∀k≥0.
This proves Claim1.
Claim 2.
Ck(φ,u0)=Ck(φ+,u0),Ck(φ,v0)=Ck(φ-,v0) for all k≥0.
We consider the homotopy(3.117)h1(t,u)=(1-t)φ+(u)+tφ(u)∀(t,u)∈[0,1]×W01,p(Ω).
Clearly u0∈Kh(t,·) for all t∈[0,1]. We will show that there exists ϱ>0, such that
(3.118)Bϱ(u0)∩Kh1(t,⋅)={u0}∀t∈[0,1],
where
(3.119)Bϱ(u0)={u∈W01,p(Ω):‖u-u0‖<ϱ}.
Arguing by contradiction, suppose that (3.118) is not true for any ϱ>0. Then we can find two sequences {tn}n≥1⊆[0,1] and {un}n≥1⊆W01,p(Ω)∖{u0}, such that
(3.120)tn⟶tin[0,1],un⟶u0inW01,p(Ω),(htn)′(un)=0∀n≥1.
For everyn≥1, we have
(3.121)A(un)=(1-tn)Nf+(un)+tnNf(un),
so
(3.122)-div⁡a(∇un(z))=(1-tn)f(z,un+(z))+tnf(z,un(z))a.e.  inΩ,un|∂Ω=0.
From Ladyzhenskaya and Uraltseva [13, p. 286], we know that we can find M9>0, such that
(3.123)‖un‖∞≤M9∀n≥1.
Then from Lieberman [14, p. 320], we infer that there exist α∈(0,1) and M10>0, such that
(3.124)un∈C01,α(Ω¯),‖un‖C01,α(Ω¯)≤M10∀n≥1.
The compactness of the embedding C01,α(Ω¯) into C01(Ω¯) and (3.120) imply that
(3.125)un⟶u0inC01(Ω¯),
so, there exists n0≥1, such that
(3.126)un∈int⁡C+∀n≥n0
(recall that u0∈intC+; see Proposition 3.5) and thus {un}n≥n0⊆Kφ+ are distinct solutions of (1.5), a contradiction.
Therefore (3.118) holds for some ϱ>0. Invoking the homotopy invariance property of critical groups, we have
(3.127)Ck(φ,u0)=Ck(φ+,u0)∀k≥0.
Similarly, we show that
(3.128)Ck(φ,v0)=Ck(φ-,v0)∀k≥0.
This proves Claim 2.
From Claims1 and 2, we have that
(3.129)Ck(φ,u0)=Ck(φ,v0)=δk,1Z∀k≥0.
Also, we have
(3.130)Ck(φ,0)=δk,0Z∀k≥0
(see Proposition 3.3) and
(3.131)Ck(φ,∞)=0∀k≥0
(see Proposition 3.7).
SinceKφ={0,u0,v0}, from (3.129), (3.130), (3.131), and the Morse relation (2.13) with t=-1, we have
(3.132)2(-1)1+(-1)0=0,
a contradiction.
Therefore, there existsy0∈Kφ, y0∉{0,u0,v0}. So, y0 solves (1.5), and by the nonlinear regularity theory, y0∈C01(Ω¯).Remark 3.9.
Even in the Hilbert space case (i.e.,p=2), our result is more general than that of Wang [1], since we go beyond the Laplace differential operator and our hypotheses on the reaction f are considerably more general.



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*Source: 918271-2012-05-15.xml*