# Existence and Multiplicity of Solutions for a Class of Anisotropic Double Phase Problems

Authors: Jie Yang; Haibo Chen; Senli Liu
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

We consider the following double phase problem with variable exponents:−div∇upx−2∇u+ax∇uqx−2∇u=λfx,uinΩ,u=0,on∂Ω. By using the mountain pass theorem, we get the existence results of weak solutions for the aforementioned problem under some assumptions. Moreover, infinitely many pairs of solutions are provided by applying the Fountain Theorem, Dual Fountain Theorem, and Krasnoselskii’s genus theory.


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## Body

## 1. Introduction and Statement of Results

In this paper, we deal with the existence and multiplicity of solutions for the following double phase problem(1)−div∇upx−2∇u+ax∇uqx−2∇u=λfx,u,inΩ,u=0,on∂Ω,Pλwhere λ>0 is a real parameter, Ω⊂ℝNN≥2 is a bounded domain with smooth boundary, p∗⋅=Np⋅/N−p⋅, p⋅, and q⋅ are Lipschitz continuous in ℝN. Moreover,
(2)q⋅p⋅<1+1N,a:Ω¯⟶0,+∞,isLipschitzcontinuousand we also assume that the nonlinearity f satisfies the following conditions:f1f:Ω×ℝ⟶ℝ is a Carathéodory function and there exists C1>0 such that
(3)fx,t≤C11+tαx−1,for all x,t∈Ω×ℝ, where α∈CΩ¯,1<q+<α−≤α+<p∗·.f2limt→0fx,t/tq+−1=0, uniformly for a.e. x∈Ω. f3limt→+∞Fx,t/tq+=+∞, uniformly for a.e. x∈Ω, where Fx,t=∫0tfx,sds.f4 There exists a constant C0>0 such that
(4)Gx,t≤Gx,s+C0,for any x∈Ω,0<t<s or s<t<0, where Gx,t=tfx,t−q+Fx,t.f4∗ There exists T0>0 such that fx,t/tq+−2t is nondecreasing in t when t≥T0 and nonincreasing in t≤−T0 for all x∈Ω.f5fx,−t=−fx,t, for all x∈Ω and t∈ℝ.Remark 1.
We point out that the conditionf4 is weaker than f4∗. It is not difficult to check that the condition f4∗ is equivalent to the following condition (see [1]): f4∗∗Gx,t is increasing in t≥T0 and decreasing in t≤−T0 for all x∈Ω.
Hence,f4∗ implies f4.Similar problems have been investigated and it is well known they have a strong physical meaning because they appear in the models of strongly anisotropic materials, see, e.g., [2, 3]. The energy functionals of the form
(5)u↦∫Ωℋx,∇uxdx,ℋx,t=tpx+axtqx,qx>px>1,a⋅>0,where the integrand H switches between two different elliptic behaviors have been intensively studied in recent years, see [2–11]. Recently, Mingione et al. have obtained the regularity theory for minimizers of (5), see, e.g., [7].Whenax=1 and λ=1, problem Pλ becomes a px,qx-Laplacian problem of the form
(6)−Δpxux−Δqxux=fx,u,inΩ,u=0,on∂Ω,where −Δpxu≔−div∇upx−2∇u. In particular, we refer to [9] where the authors proved the existence of one and three nontrivial weak solutions of (6), by the mountain pass theory and Morse theory.Ifpx=qx, then ax=1. Vetro [12] studied the following Dirichlet boundary value problem involving the px-Laplacian-like operator:
(7)−Δpxlux+uxpx−2ux=λfx,u,inΩ,u=0,on∂Ω,where
(8)−Δpxlu≔div1+∇upx1+∇u2px∇upx−2∇u,is the px-Laplacian-like. They have established the existence and multiplicity results for the problem (7) when λ is sufficiently small.In the particular case ofpx≡p,qx≡q, such problems have been recently studied in, e.g., [13–16]. The existence and multiplicity of weak solutions of problem Pλ with λ=1 has been established in Liu and Dai [13]. In [15], by using the Morse theory, Perera and Squassina obtained a nontrivial weak solution of problem Pλ. In [14], by utilizing the Nehari method, Liu and Dai obtained three ground state solutions. Usually, the authors in those references considered the nonlinearities fx,t satisfying the Ambrosetti-Rabinowitz type condition ((AR) in short): i.e., there exist L>0, θ>q, such that for ∣t∣≥L and a.e. x∈Ω,
(9)0<θFx,t≤tfx,t.Under some appropriate assumptions, one can consider a much weaker condition onfx,t
(10)limu→+∞Fx,uuq=+∞uniformlyinx.This means thatF is q-superlinear at infinity. But the (AR) condition is useful and natural to ensure the mountain pass geometry and the Palais-Smale condition ((PS) in short). So it have attracted much interest in recent literature, see for example [13, 15, 17–19] and the references therein. However, in this paper, we consider the problem Pλ in the case when the nonlinearity F is q+-superlinear at both infinity and origin (see conditions f2 and f3). These conditions are weaker than the (AR) condition. For example, Papageorgiou, Vetro, and Vetro [16] investigated the following (p,2)-equation with combined nonlinearities:
(11)−Δpux−Δux=λfx,u+gx,u,inΩ,u=0,on∂Ω,where λ>0,2<p<+∞,Ω⊂ℝN, be a bounded domain with a C2-boundary ∂Ω. Using the critical point theory, critical groups, and flow invariance arguments, the authors obtained at least five nontrivial smooth solutions of (11) when f is (p−1)-superlinear near ±∞ but does not satisfy the (AR) condition.Now, a natural question is whether the results contained in [13] can be generalized to the variable exponents px,qx case. Moreover, can we assume that the nonlinearity f satisfies a more natural and weaker q+−1-superlinear condition near ±∞ instead of the (AR) condition?Inspired by the above works, we will answer these questions. For a detailed motivation of our context and additional references, we refer to the introduction of [8, 20]. To the best of our knowledge, there are very few papers related to the existence of solutions of problem Pλ with variable exponents. This paper was motivated by the interest in applications of the variable exponent Orlicz-Sobolev spaces. Before stating our main results, we introduce some notations.


### 1.1. Notations and definitions

Throughout this paper, we define the class(12)C+Ω¯=p∈CΩ¯,px>1forallx∈Ω¯.For anyp∈C+Ω¯, we denote
(13)p+≔esssupx∈ℝNpx,p−≔essinfx∈ℝNpx,and we denote by p1≪p2 the fact that
(14)essinfx∈ℝNp2x−p1x>0.The lettersC,Ci,i=1,2,⋯, denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. The notion of weak solution for problem Pλ is that u∈W01,ℋΩ is a solution of Pλ if
(15)∫Ω∇upx−2+ax∇uqx−2∇u⋅∇vdx=∫Ωfx,uvdx,∀v∈W01,ℋΩ.It is formulated in a suitable Orlicz-Sobolev spaceW01,ℋΩ that will be introduced in Section 2. It is easy to see that solutions of Pλ correspond to the critical points of the energy functional Iλ defined by
(16)Iλu=∫Ω1px∇upx+axqx∇uqxdx−λ∫ΩFx,udx,∀u∈W01,ℋΩ,where Fx,t=∫0tfx,sdx.Now, we present the main results of this paper as follows:Theorem 2.
Supposef1−f4 are satisfied. Then problem Pλ has at least one nontrivial weak solution in W01,ℋΩ for all λ>0.Theorem 3.
Supposef1,f3−f4 are satisfied. Then there exists λ0>0 such that for all λ∈0,λ0, problem Pλ has at least one solution uλ and
(17)limλ→0+uλ=+∞.Theorem 4.
Supposef1−f5 are satisfied. Then problem Pλ has infinitely many solutions in W01,ℋΩ for all λ>0.Theorem 5.
Supposef5 and the following conditionf6f:Ω×ℝ⟶ℝ is a Carathéodory function, and there exist positive constants d0,d1 such that
(18)d0tβx−1≤fx,t≤d1tβx−1,for all x∈Ω¯and t≥0, where β∈CΩ¯ such that 1<βx<p∗x with β+<p−. Then problem Pλ has infinitely many solutions in W01,ℋΩ for all λ>0.Theorem 6.
Supposef1,f3−f5 are satisfied. Then for all λ>0, problem Pλ has infinitely many solutions unn∈ℕW01,ℋΩ such that limn→∞Iλun=∞.Theorem 7.
Supposef1,f3−f5 are satisfied. Then for all 0<λ<α/q+, problem Pλ has infinitely many solutions vnn∈ℕW01,ℋΩ such that Iλvn<0,limn→∞Iλvn=0.Remark 8.
Note that our Theorems2–7 answer the above questions. To be precise, Theorems 2, 4, 6, and 7 extend the main results of [13] to the variable exponents px,qx case. Compared with [13], the main difficulty is that since both px and qx are nonconstant functions, then Pλ has a more complicated structure, due to its nonhomogeneities and to the presence of the nonlinear term.Remark 9.
In Theorem5, we obtain infinitely many solutions by using Krasnoselskii’s genus theory. Moreover, we consider continuous functions f=fx,u satisfying the growth condition
(19)d0uβx−1≤fx,u≤d1uβx−1.The rest of this paper is organized as follows. In Section2, we state some preliminary notations and the main lemmas. In Section 3, we prove the Theorems 2 and 3. The proofs of Theorems 4–5 are given in Section 4. By using the Fountain Theorem and the Dual Fountain Theorem, infinitely many pairs of solutions are provided in Section 5.


## 1.1. Notations and definitions

Throughout this paper, we define the class(12)C+Ω¯=p∈CΩ¯,px>1forallx∈Ω¯.For anyp∈C+Ω¯, we denote
(13)p+≔esssupx∈ℝNpx,p−≔essinfx∈ℝNpx,and we denote by p1≪p2 the fact that
(14)essinfx∈ℝNp2x−p1x>0.The lettersC,Ci,i=1,2,⋯, denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. The notion of weak solution for problem Pλ is that u∈W01,ℋΩ is a solution of Pλ if
(15)∫Ω∇upx−2+ax∇uqx−2∇u⋅∇vdx=∫Ωfx,uvdx,∀v∈W01,ℋΩ.It is formulated in a suitable Orlicz-Sobolev spaceW01,ℋΩ that will be introduced in Section 2. It is easy to see that solutions of Pλ correspond to the critical points of the energy functional Iλ defined by
(16)Iλu=∫Ω1px∇upx+axqx∇uqxdx−λ∫ΩFx,udx,∀u∈W01,ℋΩ,where Fx,t=∫0tfx,sdx.Now, we present the main results of this paper as follows:Theorem 2.
Supposef1−f4 are satisfied. Then problem Pλ has at least one nontrivial weak solution in W01,ℋΩ for all λ>0.Theorem 3.
Supposef1,f3−f4 are satisfied. Then there exists λ0>0 such that for all λ∈0,λ0, problem Pλ has at least one solution uλ and
(17)limλ→0+uλ=+∞.Theorem 4.
Supposef1−f5 are satisfied. Then problem Pλ has infinitely many solutions in W01,ℋΩ for all λ>0.Theorem 5.
Supposef5 and the following conditionf6f:Ω×ℝ⟶ℝ is a Carathéodory function, and there exist positive constants d0,d1 such that
(18)d0tβx−1≤fx,t≤d1tβx−1,for all x∈Ω¯and t≥0, where β∈CΩ¯ such that 1<βx<p∗x with β+<p−. Then problem Pλ has infinitely many solutions in W01,ℋΩ for all λ>0.Theorem 6.
Supposef1,f3−f5 are satisfied. Then for all λ>0, problem Pλ has infinitely many solutions unn∈ℕW01,ℋΩ such that limn→∞Iλun=∞.Theorem 7.
Supposef1,f3−f5 are satisfied. Then for all 0<λ<α/q+, problem Pλ has infinitely many solutions vnn∈ℕW01,ℋΩ such that Iλvn<0,limn→∞Iλvn=0.Remark 8.
Note that our Theorems2–7 answer the above questions. To be precise, Theorems 2, 4, 6, and 7 extend the main results of [13] to the variable exponents px,qx case. Compared with [13], the main difficulty is that since both px and qx are nonconstant functions, then Pλ has a more complicated structure, due to its nonhomogeneities and to the presence of the nonlinear term.Remark 9.
In Theorem5, we obtain infinitely many solutions by using Krasnoselskii’s genus theory. Moreover, we consider continuous functions f=fx,u satisfying the growth condition
(19)d0uβx−1≤fx,u≤d1uβx−1.The rest of this paper is organized as follows. In Section2, we state some preliminary notations and the main lemmas. In Section 3, we prove the Theorems 2 and 3. The proofs of Theorems 4–5 are given in Section 4. By using the Fountain Theorem and the Dual Fountain Theorem, infinitely many pairs of solutions are provided in Section 5.


## 2. Preliminaries

In order to discuss the problemPλ, we need some theories on generalized Orlicz spaces and Sobolev spaces. For more details, we refer to the references [20–23]. The variable exponent Lebesgue space LpxΩ is defined by
(20)Lp·Ω=uisameasurablerealvaluedfunction∫Ωuxpxdx<+∞,endowed with the Luxemburg norm
(21)up⋅=infλ>0:∫Ωuxλpxdx≤1.Note that, ifp is a constant function, the Luxemburg norm up⋅ coincide with the standard norm up of the Lebesgue space LpΩ. Then, (LpxΩ,up⋅) becomes a Banach space, and we call it the variable exponent Lebesgue space. It is easy to check that the embedding Lp2xΩ↪Lp1xΩ is continuous, where 0<∣Ω∣<∞ and p1,p2 are variable exponents such that p1≤p2 in Ω.The following property of spaces with variable exponent is essentially due to Fan and Zhao [24].Lemma 10.
The spaceLp⋅Ω,⋅p⋅ is a separable, uniformly convex Banach space, and its dual space is Lp′⋅Ω where 1/px+1/p′x=1. For any u∈Lp⋅Ω and v∈Lp′⋅Ω, we have
(22)∫Ωuvdx≤1p−+1p′−up⋅vp′⋅.The Musielak-Orlicz spaceLHΩ is defined by
(23)LℋΩ=u:Ω⟶ℝmeasurable:∫Ωℋx,udx<+∞,endowed with the norm
(24)uℋ=infλ>0:ρℋuλ≤1,where ℋ is defined in (5). The space LHΩ is a separable, uniformly convex, and reflexive Banach space. We denote by Laq⋅Ω the space of all measurable functions u:Ω⟶ℝ with the seminorm
(25)up⋅,a≔∫Ωaxuqxdx1/qx<∞.It is easy to check that the embeddings(26)Lq∙Ω↪LHΩ↪Lp∙Ω∩Laq∙Ω,are continuous. Since ρℋu/uℋ=1 whenever u≠0, we have
(27)minuℋpx,uℋqx≤up⋅px+up⋅,apx≤maxuℋpx,uℋqx,∀u∈LℋΩ.The related Sobolev spaceW1,HΩ is defined by
(28)W1,ℋΩ≔u∈LℋΩ:∇u∈LℋΩ,equipped with the norm
(29)u=uℋ+∇uℋ,where ∇uℋ=∇uℋ. The completion of C0∞Ω in W1,HΩ is denoted by W01,HΩ and it can be equivalently renormed by
(30)u≔∇uℋ,via a Poincaré-type inequality, cf ([6], Proposition 2.18(iv)), under assumption (2). The spaces W1,HΩ and W01,HΩ are uniformly convex, and hence reflexive, Banach space. By (27), We have
(31)minupx,uqx≤∇up⋅px+∇up⋅,apx≤maxupx,uqx,∀u∈W01,ℋΩ.We point out that ifr∈C+Ω¯ and rx≤p∗x for all x∈Ω¯, then W1,HΩ↪Lr∙Ω is continuous. This embedding is compact if
(32)infx∈Ωp∗x−rx>0.Let us now defineJ⋅:W01,ℋΩ⟶ℝ as
(33)Ju=∫Ω1px∇upx+axqx∇uqxdx,and we denote the derivative operator by A, that is A=J′:W01,ℋΩ⟶W01,ℋΩ∗, with
(34)Au,v=∫Ω∇upx−2+ax∇uqx−2∇u⋅∇vdx,u,v∈W01,ℋΩ.Here,W01,ℋΩ∗ denotes the dual space of W01,ℋΩ, and ⋅,⋅ denotes the paring between W01,ℋΩ and W01,ℋΩ∗. In the following lemma, we summarize some properties of A, useful to study our problem. When px≡p,qx≡q, we refer to ([13], Proposition 3.1).Lemma 11 (see [19], Lemma 3.4).
Under the condition (2), A is a mapping of type S+, that is, if un⇀u in W01,ℋΩ and limsupn→+∞Aun−Au,un−u≤0,, then un⟶u in W01,ℋΩ.Lemma 12 (see [19], Lemma 3.2).
Under the conditionf1, Iλ is well defined on W01,ℋΩ, and Iλ∈C1W01,HΩ,ℝ with Fréchet derivate given by
(35)Iλ′u,v=∫Ω∇upx−2+ax∇uqx−2∇u·∇vdx−λ∫Ωfx,uvdx,u,v∈W01,HΩ.
Firstly, we show the functionalIλ satisfies the Cc condition.Lemma 13.
If hypothesesf1,f3, and f4 hold, then Iλ satisfies the Cc condition.Proof.
For everyc∈ℝ, let un⊂W01,HΩ be a Cc−sequence, that is,
(36)Iλun⟶c,and∥I′λun∥W01,HΩ∗1+∥un∥⟶0,asn⟶∞.We claim that {un} is bounded in W01,ℋΩ. In fact, suppose by contradiction that ∥un∥⟶+∞, as n⟶∞. Let vn=un/∥un∥,n≥1. Up to a subsequence, we may assume that
(37)vn⟶v,a.e.inΩ,vn⇀v,weaklyinW01,HΩ,vn⟶v,stronglyinLq+Ω,vn⟶v,stronglyinLαxΩ.We know thatv satisfies the following alternative: v=0 or v≠0. In what follows, we will show that under the condition ∥un∥⟶+∞, v satisfies neither v=0 nor v≠0. This is a contradiction. Thus, un is bounded.Ifv=0, then vn⟶0 a.e. x∈Ω, as n⟶∞. Since Iλtun is continuous in t∈0,1, for each n, there exists tn∈0,1n=1,2,⋯ such that
(38)Iλtnun=maxt∈0,1Iλtun.It is easily seen thattn>0 and Iλtnun≥c>0=Iλ0=Iλ0un. If tn<1, then d/dtIλtunt=tn=0, which implies
(39)Iλ′tnun,tnun=0.Moreover, iftn=1, then, from(36) we have Iλ′un,un=on1. So, we always have
(40)Iλ′tnun,tnun=on1.Letγk be a sequence of positive real numbers such that γk>1 for any k and limk→+∞γk=+∞. Then ∥γkvn∥=γk>1 for any k and n. Fix k, using f1, (37), and the Lebesgue dominated convergence theorem we deduce that
(41)limn→∞∫ΩFx,γkvndx=0.Recall that∥un∥⟶+∞ as n⟶∞. So, we have ∥un∥>γk or 0<γk/∥un∥<1 for n large enough. Hence, from (31) and (38), we deduce that
(42)Iλtnun≥Iλγk∥un∥un=Iλγkvn=∫Ωγkpxpx∇vnpx+γkqxqxax∇vnqx−λFx,γkvndx≥γkp−q+−λ∫ΩFx,γkvndx,for any n large enough. By combing this inequality with (41), as n,k⟶+∞, we have
(43)limsupn→∞Iλtnun=+∞.On the other hand, using conditionf4 and (40), for all n large enough, we obtain
(44)Iλtnun=Iλtnun−1q+Iλ′tnun,tnun+o1=∫Ω1px−1q+∇tnun∣pxdx+∫Ω1qx−1q+ax∇tnunqxdx+λ∫Ω1q+fx,tnuntnun−Fx,tnundx≤Iλun−1q+Iλ′un,un+λC0Ωq+⟶c+λC0∣Ω∣q+,asn⟶∞.From (43) and (44), we obtain a contradiction. This shows that v≠0, and thus,
(45)vnx⟶vx≠0a.e.inΩ.LetΩ≠≔x∈Ω:vx≠0. It implies that
(46)∣unx∣⟶+∞,inΩ≠,asn⟶∞.Using conditionf3, we obtain
(47)limn→+∞Fx,unx∥unx∥q+=limn→+∞Fx,unxunxq+unxq+∥unx∥q+=limn→+∞Fx,unxunxq+vnxq+=+∞,x∈Ω≠.Also byf1 and f3, we can get a constant C2>0 such that
(48)Fx,t≥−C2,∀x,t∈Ω¯×ℝ.Thus, we get(49)Fx,un+C2∥un∥q+≥0.From (31), we see that
(50)c=Iλunx+on1=∫Ω1px∇unpx+axqx∇unqx−λFx,undx+on1≥1q+∥un∥p−−∫ΩλFx,undx+on1,which implies
(51)∫ΩFx,undx≥1λq+∥un∥p−−cλ+on1⟶+∞,asn⟶∞.Similarly, from (31), we also get
(52)c=Iλunx+on1≤1p−∥un∥q+−∫ΩλFx,undx+on1,which implies
(53)∥un∥q+≥p−c+λp−∫ΩFx,undx−on1>0,for n large enough.We claim that∣Ω≠∣=0. Indeed, suppose by contradiction ∣Ω≠∣≠0, then by (47)–(53) and Fatou’s lemma, we obtain
(54)+∞=∫Ω≠liminfn→+∞Fx,unxunxq+vnxq++C2∥un∥q+≤liminfn→+∞∫Ω≠Fx,unxunxq+vnxq++C2∥un∥q+=liminfn→+∞∫Ω≠Fx,unx∥unx∥q+≤liminfn→+∞∫ΩFx,unxdxp−c+λp−∫ΩFx,undx−on1=1λp−,which yields a contradiction. Therefore the sequence un is bounded in W01,HΩ. Thus, there is a subsequence (which we still denote by un) that converges weakly to some u∈W01,HΩ and strongly in Lα·Ω. It is easy to check from (f1) and Hölder’s inequality that
(55)∫Ωfx,unun−udx≤C∥1+unαx−1∥α′·∥un−u∥α·⟶0.Then(56)Aun,un−u=I′un,un−u+λ∫Ωfx,unun−udx⟶0.Soun⟶u follows from Lemma 11.


## 3. Proofs of Theorems2 and 3

First, we will show the functionalIλ satisfies the mountain pass geometry [25].Lemma 14.
Assume hypothesesf1−f3 hold. Then the functional Iλ satisfies the following properties:
(i)
There existρ,δ>0 such that Iλu≥δ for any u∈W01,HΩ with ∥u∥=ρ
(ii)
There exists aη∈W01,HΩ\Bρ such that Iλη≤0.Proof.
Let us check (i). For anyu∈W01,HΩ\0 and ε>0 small, it follows from f1−f2 that there exists Cε>0 such that
(57)Fx,t≤εtq++Cεtαx,forallx,t∈Ω×ℝN.
Thus, foru∈W01,HΩ and ∥u∥≤1, we have
(58)Iλu=∫Ω1px∇upx+axqx∇uqx−λFx,udx,≥∫Ω1px∇upx+axqx∇uqxdx−λ∫Ωεuq++Cεuαxdx,≥1q+∥u∥q+−λC3ε∥u∥q+−λCεC4∥u∥α−,by the Sobolev embedding W01,HΩ↪Lq+Ω and W01,HΩ↪Lα·Ω. Since q+<α− and ε arbitrarily small, there exist ρ>0 and δ>0 such that Iλu≥δ>0 for ∥u∥=ρ. Hence item (i) holds.
Let us check (ii). Fromf3, for any M>0, we can choose a constant C5>0 such that
(59)Fx,t≥Mtq+−C5,∀x,t∈Ω×ℝ.
Then, forω∈W01,HΩ and t>0, we deduce that
(60)limt→+∞Itwtq+≤limt→+∞∫Ω1/px∇tw∣px+ax/qx∇twqxdx−λ∫ΩMtwq+−C5dxtq+,≤limt→+∞1tq+∫Ωtpxpx∇ωpx+axtqxqx∇ωqx−tq+λMωq++λC5dx≤∫Ω1px∇ωpx+axqx∇ωqx−λMωq+dx.
IfM is large enough such that
(61)∫Ω1px∇ωpx+axqx∇ωqx−λMωq+dx<0,conclusion (ii) follows.Proof of Theorem 2.
Since the functionalIλ has the mountain pass geometry and satisfies the Cc condition, the mountain pass theorem [25] gives that there exists a critical point u∈W01,HΩ. Moreover, Iu=c≥α>0=I0, so u is a nontrivial solution.Lemma 15.
Assumef1 holds. Then there exist positive constants mλ and ρλ such that limλ→0+mλ=+∞ and Iλ≥mλ>0 when ∥u∥=ρλ.Proof.
Letu∈W01,HΩ with ∥u∥>1. It follows from f1 that there exists C6>0 such that
(62)Fx,t≤C6tαx+1,for all x,t∈Ω×ℝ,q+<αx<p∗−. Hence, we obtain
(63)Iλu≥∫Ω1px∇upx+axqx∇uqxdx−λC6∫Ωuαx+1dx≥1q+∥u∥p−−λC7uα+−λC6Ω.
Letρλ=λ−s where s∈0,1/α+−p+. Hence, we get ρλ>1 for λ small enough. Therefore, substituting ∥u∥=ρλ=λ−s in (63), we see that
(64)Iλu≥1q+λ−sp−−C7λ1−sα+−λC6Ω.
Let us definemλ=1/q+λ−sp−−C7λ1−sα+−λC6Ω.From s∈0,1/α+−p+, we get that there exist λ0 small enough such that mλ>0 for all λ∈0,λ0 and mλ⟶+∞ as λ⟶0+.Proof of Theorem 3.
By Lemma13, Iλ satisfies the Cc condition. Now in view of Lemma 13 and Lemma 15 and Lemma 14(ii) we can apply the mountain pass theorem to obtain a nontrivial critical point uλ for Iλ such that
(65)Iλuλ=c≥mλ.
On the other hand, from (62), we have
(66)Iλu≤∫Ω1px∇upx+axqx∇uqxdx+λC6∫Ωuα++1dx,≤1p−max∥uλ∥p−,∥uλ∥q++λC8maxuλα+uλα−+λC6Ω.
Taking the limitλ⟶0+ in (66) and using Lemma 15, one has limλ→0+∥uλ∥=+∞.


## 4. Proofs of Theorems4 and 5

Lemma 16.
Assume the hypothesesf1−f3 hold. Then the functional Iλ satisfies the following properties:
(i)
There exist constantsρ,δ>0, such that Iλu≥δ for any u∈W01,HΩ with ∥u∥=ρ
(ii)
For each finite dimensional subspaceX~⊂W01,HΩ, there exists an R=RX~ such that Iλ≤0,onX~\BRX~.Proof.
As in the proof of Lemma14, it is immediate to see that the case (i) is true. Let e∈X~ and ∥e∥=1 be fixed. From (59), we obtain
(67)Iλte=∫Ω1px∇tepx+axqx∇teqx−λFx,tedx≤tq+p−−λMC9tq++λC5Ω,for all norms on X~ are equivalent. Then, we can choose M large enough such that 1/p−−λMC9<0. Therefore, we see that Iλte⟶−∞, as n⟶∞, and the step is proved by taking v0=t0e with t0>R large enough.Proof of Theorem 4.
According to our assumption (f5), Iλ is an even functional. By the Lemma 13, Iλ satisfies the Cc condition. Together with the Lemma 16, we can apply a Z2 version of the mountain pass theorem (see [25], Theorem 9.12) to obtain an unbounded sequence of weak solutions of problem (Pλ).
We finalize the section presenting a relation between the genus ofK and the number of solutions of the problem Pλ, where K is a k-dimensional linear subspace K⊂C0∞Ω of W01,HΩ. We invoke Clark’s Theorem in [25], Theorem 9.1. The next result is a compactness result on problem Pλ which we will use later.Lemma 17.
Assume that conditionf6 holds, then
(i)
Iλ is bounded from below
(ii)
Iλ satisfies the (PS) condition.Proof.
(i) Usingf6, and for ∥u∥>1,λ>0, we obtain
(68)Iλu≥∫Ω1px∇upx+axqx∇uqxdx−λd1β−∫Ωuβxdx,≥1q+∥u∥p−−C10∥u∥β+.
Hence,Iλ is coercive following immediately from the above expression and β+<p−. Therefore, Iλ is bounded from below.
(ii) Supposeun is a PSc sequence for Iλ. Thus Iλun⟶c and Iλ′un⟶0 in W01,HΩ∗ as n⟶+∞. It follows from (i) that un is bounded in W01,HΩ. Up to a subsequence, we may assume that
(69)un⟶u,a.e.inΩ,un⇀u,weaklyinW01,HΩ,un⟶u,stronglyinLβ·Ω.
SinceIλ′un⟶0 and un−u⇀0 in W01,HΩ, (see [26], Proposition 3.5), we get that
(70)limn→+∞Iλ′un,un−u=0.
It is easy to check from (f6) and Hölder’s inequality that
(71)∫Ωfx,unun−udx≤C11∥unβx−1∥β′·∥un−u∥β·⟶0,asn⟶∞,where β′·=β·/β·−1. Then
(72)Aun,un−u=I′un,un−u+λ∫Ωfx,unun−udx⟶0,asn⟶∞.
Soun→u follows from Lemma 11.Proof of Theorem 5.
ConsiderK is a k-dimensional linear subspace K⊂C0∞Ω of W01,HΩ. We claim IλK<0 if ∥u∥≤r<1 is sufficiently small. Indeed, by the equivalence of norms on K, there exists a constant C12>0 such that C12∥u∥β+≤∫Ωuβxdx for u∈K with ∥u∥≤1. Therefore, by f6,
(73)Iλu≤∫Ω1px∇upx+axqx∇uqxdx−λd0β−∫Ωuβxdx≤1p−∥u∥p−−λC13∥u∥β+≤∥u∥β+1p−∥u∥p−−β+−λC13,for u∈K with ∥u∥<1. If r∈0,1 is small enough, we have that
(74)1p−rp−−β+−λC13<0.
The last inequality showsIλK<0 for all u∈Srk=u∈K:∥u∥=r. It is clear that K is isomorphic to ℝk and Srk is homeomorphic to Sk−1 in ℝk. Hence, we obtain γSrk=k. In the proof of Lemma 17, it was already established that Iλ∈C1X,ℝ is bounded from below, satisfies the (PS) condition, and Iλ0=0. Clearly, f5 implies Iλ is even. Consequently, by Clark’s Theorem in [25] (Theorem 9.1), Iλ possesses at least k distinct pairs of nontrivial solutions. Since k is arbitrary, we obtain infinitely many nontrivial solutions.


## 5. Proofs of Theorems 6 and 7

In this section, we will show that (Pλ) has infinitely many pairs of solutions by using the Fountain Theorem and Dual Fountain Theorem. Firstly, we need to recall some preliminary results. Since W01,HΩ is a reflexive and separable Banach space, there are ej⊂W01,HΩ and ej∗⊂W01,HΩ∗ such that
(75)W01,HΩ=spanej:j=1,2⋯¯,W01,HΩ∗=spanej∗:j=1,2⋯,¯ej∗,ej=1,i=j,0,i≠j.Then, we define(76)Xj=spanej,Yk=⊕j=1kXj,Zk=⊕j=k∞Xj¯.We will apply the following Fountain Theorem ([25], Theorem 3.6).Lemma 18.
Assume thatX is a Banach space, and let φ∈C1X,R be an even functional. If, for every k∈ℕ, there exists ρk>rk>0 such that
(77)A1bk≔infu∈Zk∥u∥=rkφu⟶+∞,k⟶+∞,A2ak≔maxu∈Yk∥u∥=ρkφu≤0,A3φsatisfiestheCcconditionforeveryc>0.
Thenφ has an unbounded sequence of critical values.To prove Theorems6 and 7, the following lemma is needed.Lemma 19.
Assume thatαx∈C+Ω¯,q+<αx<p∗−, for any x∈Ω¯. Let
(78)βk=sup∥u∥=1u∈Zk∥u∥Lα·,then limk→+∞βk=0.Proof.
Obviously,0<βk+1≤βk and so βk⟶β≥0. Let uk∈Zk satisfy
(79)∥uk∥=1,0≤βk−∥uk∥Lαx<1k.
Then, there exists a subsequence ofuk (which we still denote by uk) such that uk⇀u, and
(80)ej∗,u=limk→+∞ej∗,uk=0,j=1,2,⋯,which implies u=0, and thus, uk⇀0. Since W01,HΩ↪↪Lα·Ω, then uk⟶0 in Lα·Ω. Hence, we get limk→+∞βk=0.Proof of Theorem 6.
LetX=W01,HΩ. According to fx,−t=−fx,t, Iλ is an even functional. As the proof of Lemma 13, it follows from f1,f3, and (f4) that Iλ satisfies the Cc condition. For every k∈ℕ, we shall prove that there exist ρk>rk>0 such that
(81)A1bk≔infu∈Zk∥u∥=rkIλu⟶+∞,k⟶+∞,A2ak≔maxu∈Yk∥u∥=ρkIλu≤0,
We first show that (A1) holds. For any u∈Zk, we choose ∥u∥=rk=2q+C6λβkα+1/p−−α+. From Lemma 19 and p−<α+, we see that rk⟶+∞ as k⟶+∞. As before, we also have from (62) that
(82)Iλu≥1q+∥u∥p−−C6λ−λC6Ω,∥u∥α·≤1,1q+∥u∥p−−C6λβkα+∥u∥α+−λC6Ω,∥u∥α·≥1,≥12q+rkp−−λC14Ω,which implies that bk⟶+∞,k⟶+∞.
Afterwards, we demonstrate that (A2) holds. Let ϕ∈Yk and ∥ϕ∥=1,t>1. From (59), we obtain
(83)Iλtϕ=∫Ω1px∇tϕpx+axqx∇tϕqx−Fx,tϕdx≤tq+p−−λMC15tq++λC5Ω,for all norms on Yk are equivalent. Then, we can choose M large enough such that 1/p−−λMC15<0. Therefore, we see that Iλtϕ⟶−∞, as t⟶+∞. Hence, there exists t1>rk>1 large enough such that Iλt1ϕ≤0. Therefore, let ρk=t1, we obtain that ak≔maxu∈Yk∥u∥=ρkIλu≤0.For the proof of Theorem7, we need the following definitions and results.Definition 20.
LetX be a separable and reflexive Banach space, I∈C1X,ℝ,c∈ℝ. We say that I satisfies the Cc∗ condition (with respect to (Yn)), if any sequence unn∈ℕ⊂X for which un∈Yn, for any n∈ℕ, Iun⟶c and ∥I∣Yn′un∥X∗1+∥un∥⟶0, as n⟶∞, contains a subsequence converging to a critical point of I.We are now ready to prove the Theorem7.Proof of Theorem 7.
According to the Dual Fountain Theorem ([25], Theorem 3.18), it suffices to prove that for every k≥k0, there exist ρk>rk>0 such that
(84)B1ak≔maxu∈Yk∥u∥=rkIλu<0,B2bk≔infu∈Zk∥u∥=ρkIλu≥0,B3dk≔infu∈Zk∥u∥≤ρkIλu⟶0,k⟶+∞.B4IλsatisfiestheCc∗conditionforeveryc∈ℝ.
Firstly, we show that (B1) holds. Let ϕ∈Yk and ∥ϕ∥=1,t>1. Then similar to the proof of (A2), we see that
(85)Iλtϕ=∫Ω1px∇tϕpx+axqx∇tϕqx−Fx,tϕdx≤tq+p−−λMC15tq++λC5Ω,for all norms on Yk are equivalent. Then, we can choose M large enough such that 1/p−−λMC15<0. Therefore, we see that Iλtϕ⟶−∞, as t⟶+∞. Hence, there exists t2>1 large enough such that Iλt2ϕ<0. Therefore, let rk=t2, we obtain that
(86)ak≔maxu∈Yk∥u∥=rkIλu<0.
We show thatB2 holds. As we have done in the proof of Theorem 6, For any u∈Zk, choosing ∥u∥=ρk=2q+C6λβkα+1/p−−α+. From Lemma 19 and p−<α+, we see that ρk⟶+∞ as k⟶+∞. As before, we also have from (88) that
(87)Iλu≥1q+∥u∥p−−C6λ−λC6Ω,∥u∥α·≤1,1q+∥u∥p−−C6λβkα+∥u∥α+−λC6Ω,∥u∥α·≥1,≥12q+ρkp−−λC16Ω,which implies that there exists k0∈ℕ, for all k≥k0 choosing ρk>rk>0 such that bk≥0.B3 First from Yk∩Zk≠∅ and 0<rk<ρk, we observe that
(88)dk≔infu∈Zk∥u∥≤ρkIλu≤ak≔maxu∈Yk∥u∥=rkIλu<0.
Byf1, there exists C17>0 such that
(89)∣Fx,t∣≤C17∣t∣+tαxfor all x,t∈Ω×ℝ,q+<αx<p∗−. Now we define the function Ψ1,Ψ2:X⟶ℝ by
(90)Ψ1u=∫ΩλC17uαxdx,Ψ2u=∫ΩλC17∣u∣dx.
By the definition ofΨ1,Ψ2, we have Ψi0=0,i=1,2, and they are weakly-strongly continuous. Consider
(91)ξk=supu∈Zk,∥u∥≤1Ψ1u,ζk=supu∈Zk,∥u∥≤1Ψ2u.
From the compact embeddingW01,HΩ↪Lα·Ω and Lemma 19, we have
(92)limk→+∞ξk=limk→+∞ζk=0.
Letω∈Zk and ∥ω∥=1,0<t<ρk. Then, from (89) and (90), we obtain
(93)Iλtω=∫Ω1px∇tωpx+axqx∇tωqx−λFx,tωdx≥−λ∫ΩFx,tωdx≥−Ψ1tω−Ψ2tω≥−ρkα+Ψ1ω−ρkΨ2ω≥−ρkα+ξk−ρkζk.
Passing the limit in the above inequality, ask⟶+∞, we achieve that
(94)limk→+∞dk≥0,which, together with (88), implies that limk→+∞dk=0.
B4 Let un be any sequence in W01,HΩ such that
(95)un∈Yn,Iλun⟶c>0,∥Iλ∣Yn′un∥1+∥un∥⟶0,asn⟶∞
Then similar to the proof of Lemma13, we see that un is bounded in W01,HΩ. Thus, there is a subsequence (which we denote by unk) that converges weakly to some u∈W01,HΩ and strongly in Lα·Ω. It is easy to check from (f1) and Hölder’s inequality that
(96)∫Ωfx,unkunk−udx≤C∥1+unkαx−1∥α′·∥unk−u∥α·⟶0.Claim 21.
limk→+∞Iλ′unk,unk−u=0.If Claim21 holds true, then
(97)Aunk,unk−u=I′unk,unk−u+λ∫Ωfx,unkunk−udx⟶0.Sounk⟶u follows from Lemma 11. Hence, Iλ satisfies the Cc∗ condition. In order to prove Claim 21, we invoke W01,HΩ=∪nYn¯=spanen:n=1,2,⋯¯ to choose vn∈Yn such that vn⟶u strongly in W01,HΩ. Since Iλ∣Ynk′unk⟶0 and unk−vnk⇀0 in Ynk, (see [26], Proposition 3.5), we get that
(98)limk→+∞Iλ′unk,unk−vnk=0.Hence, we obtain(99)limk→+∞Iλ′unk,unk−u=limk→+∞Iλ′unk,unk−vnk+limk→+∞Iλ′unk,vnk−u=0.Therefore, the Claim holds true and we conclude thatIλ′unk⟶Iλ′u as k⟶+∞. We next show that Iλ′u=0. To see this, taking ωj∈Yj, we have
(100)Iλ′u,ωj=Iλ′u−Iλ′unk,ωj+Iλ′unk,ωj=Iλ′u−Iλ′unk,ωj+IλYnk′unk,ωj.We pass limit in the right side of (100) as k⟶+∞ to obtain
(101)Iλ′u,ωj=0,forallωj∈Yj.Therefore,Iλ satisfies the Cc∗ condition for every c∈ℝ. The proof is complete.



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*Source: 8237492-2020-03-23.xml*