# Existence of Solutions for thep(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent

Authors: Yu Mei; Fu Yongqiang; Li Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with thep(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in W01,p(x)(Ω) space is established, then by applying it we obtain the existence of solutions for the following p(x)-Laplacian problem: -div(|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u),x∈Ω,u=0,x∈∂Ω, where Ω⊂ℝN is a bounded domain, 0∈Ω, 1<p-≤p(x)≤p+<N, and f(x,u) satisfies p(x)-growth conditions.


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## Body

## 1. Introduction

In this paper we are concerned with the followingp(x)-Laplacian problem:
(1.1)-div⁡(|∇u|p(x)-2∇u)+|u|p(x)-2u=h(x)|u|ps*(x)-2u|x|s(x)+f(x,u),x∈Ω,u=0,x∈∂Ω,
where 0∈Ω⊂ℝN is a bounded domain, p(x) is Lipschitz continuous, radially symmetric on Ω¯, and 1<p-≤p(x)≤p+<N. s(x) is Lipschitz continuous, radially symmetric on Ω¯ and 0≤s(x)≪p(x).  ps*(x)=((N-s(x))/(N-p(x)))p(x) is the critical Sobolev-Hardy exponent, and p0*(x)=Np(x)/(N-p(x))=p*(x) is the critical Sobolev exponent. Throughout this paper we assume the following:(F-1)
f(x,t) satisfies the Carathéodory condition.(F-2)
|f(x,t)|≤c1+c2|t|q(x)-1, q:Ω¯→ℝ is measurable and satisfies p(x)≪q(x)≪ps*(x) or 1<q-≤q(x)≪p(x), for any x∈Ω¯.(F-3)
f(x,t)=f(|x|,t), for any (x,t)∈Ω×ℝ.(F-4)
f(x,t)=-f(x,-t), for any (x,t)∈Ω×ℝ.(F-5)
h(x)∈C(Ω¯), h(x)=h(|x|)>0 for any 0≠x∈Ω and h(0)=0.In this paper, we mainly consider the singularity, that is,limx→0h(x)·(1/|x|s(x))=∞. For example, let h(x)=1/|ln|x∥ for x≠0; h(x)=0 for x=0; s0=infx∈Ω¯s(x)>0. It is easy to get limx→0(1/|ln|x∥)·(1/|x|s(x))=∞.Here we explain some notations employed in this paper: LetP(Ω) be the set of all Lebesgue measurable functions p:Ω→(1,∞). Forallp(x)∈P(Ω), we denote p+=supx∈Ω¯p(x), p-=infx∈Ω¯p(x), ps*+=supx∈Ω¯ps*(x), ps*-=infx∈Ω¯ps*(x) and denote by p1(x)≪p2(x) the fact that inf{p2(x)-p1(x)}>0. Denote by ci, C, and ki the generic positive constants. Denote by |Ω| the Lebesgue measure of Ω.Whenp(x)≡p is a constant function, the p-Laplacian problem related to Sobolev-Hardy inequality had been studied by many authors, either is the bounded domain or in the whole space ℝN, see, for example, [1–4]. In recent years, along with variable Sobolev spaces Lp(x)(Ω) and Wm,p(x)(Ω) being used, there are a lot of studies on p(x)-Laplacian problems, see [5–8], and the theory on problems with p(x)-growth conditions has important applications in nonlinear elastic mechanics and electrorheological fluids, see [9–12]. In [13], Fu discussed the existence of solutions for a class of p(x)-Laplacian equation with critical growth by establishing a principle of concentration compactness. The method employed in this paper is a extension of the argument in [13, 14].This paper is organized as follows: in Section2 we deal with some preliminary materials and technical results; in Section 3 we give the proof of a principle of concentration compactness; in Section 4 we study the problem of p(x)-Laplacian equation with the critical Sobolev-Hardy exponent.


## 2. Preliminaries

In this section we first recall some facts on variable exponent Lebesgue spaceLp(x)(Ω) and variable exponent Sobolev space W1,p(x)(Ω), where Ω⊂ℝN is an open set, see [15–19] for the details.Letp(x)∈P(Ω) and
(2.1)‖u‖p=inf⁡{λ>0:∫Ω‍|uλ|p(x)dx≤1}.
The variable exponent Lebesgue space Lp(x)(Ω) is the class of functions u such that ∫Ω‍|u(x)|p(x)dx<∞. Lp(x)(Ω) is a Banach space endowed with the norm (2.1).For a givenp(x)∈P(Ω), we define the conjugate function p′(x) as:
(2.2)p′=p(x)p(x)-1.Theorem 2.1.
Letp(x)∈P(Ω). Then the inequality
(2.3)∫Ω‍|f(x)⋅g(x)|dx≤2‖f‖p‖g‖p′
holds for every f∈Lp(x)(Ω) and g∈Lp′(x)(Ω).Theorem 2.2.
Suppose thatp(x) satisfies
(2.4)1<p-≤p+<∞.
Let measΩ<∞, p1(x), p2(x)∈P(Ω), then the necessary and sufficient condition for Lp2(x)(Ω)⊂Lp1(x)(Ω) is that for almost all x∈Ω we have p1(x)≤p2(x), and in this case, the imbedding is continuous.Theorem 2.3.
Suppose thatp(x) satisfies (2.4). Let ρ(u)=∫Ω‍|u(x)|p(x)dx. If u,uk∈Lp(x)(Ω), then (1)
∥u∥p<1(=1;>1) if and only if ρ(u)<1(=1;>1).(2)
If∥u∥p>1, then ∥u∥pp-≤ρ(u)≤∥u∥pp+.(3)
If∥u∥p<1, then ∥u∥pp+≤ρ(u)≤∥u∥pp-.(4)
limk→∞∥uk∥p=0 if and only if limk→∞ρ(uk)=0.(5)
∥uk∥p→∞ if and only if ρ(uk)→∞.We assume thatk is a given positive integer.Given a multi-indexα=(α1,…,αn)∈ℕn, we set |α|=α1+⋯+αn and Dα=D1α1⋯Dnαn, where Di=∂/∂xi is the generalized derivative operator.The generalized Sobolev spaceWk,p(x)(Ω) is the class of functions f on Ω such that Dαf∈Lp(x) for every multi-index α with |α|≤k. Wk,p(x)(Ω) is endowed with the norm
(2.5)‖f‖k,p=∑|α|≤k‍‖Dαf‖p.
By W0k,p(x)(Ω) we denote the subspace of Wk,p(x)(Ω) which is the closure of C0∞(Ω) with respect to the norm (2.5).In this paper we use the following equivalent norm ofW1,p(x)(Ω):
(2.6)‖u‖1,p=inf⁡{λ>0:∫Ω‍|∇uλ|p(x)+|uλ|p(x)dx≤1}.
Then we have the inequality (1/2)(∥∇u∥p+∥u∥p)≤∥u∥1,p≤2(∥∇u∥p+∥u∥p).Theorem 2.4.
The spacesWk,p(x)(Ω) and W0k,p(x)(Ω) are separable reflexive Banach spaces if p(x) satisfies (2.4).Theorem 2.5.
Suppose thatp(x) satisfies (2.4). Let φ(u)=∫Ω‍|∇u(x)|p(x)+|u(x)|p(x)dx. If u,uk∈W1,p(x)(Ω), then(1)
∥u∥1,p<1(=1;>1) if and only if φ(u)<1(=1;>1).(2)
If∥u∥1,p>1, then ∥u∥1,pp-≤φ(u)≤∥u∥1,pp+.(3)
If∥u∥1,p<1, then ∥u∥1,pp+≤φ(u)≤∥u∥1,pp-.(4)
limk→∞∥uk∥1,p=0 if and only if limk→∞φ(uk)=0.(5)
∥uk∥1,p→∞ if and only if φ(uk)→∞.Theorem 2.6.
LetΩ be a bounded in ℝN, p∈C(Ω¯) and satisfies (2.4). Then for any measurable function q(x) with 1≤q(x)≪p*(x), there is a compact embedding W1,p(x)(Ω)↪Lq(x)(Ω).Theorem 2.7.
Ifp:Ω¯→R is Lipschitz continuous and satisfies (2.4), then for any measurable function q(x) with p(x)≤q(x)≤p*(x), there is a continuous embedding W1,p(x)(Ω)↪Lq(x)(Ω).
Next let us consider the weighted variable exponent Lebesgue space. Leta(x)∈P(Ω) and a(x)>0 for x∈Ω. Define
(2.7)La(x)p(x)(Ω)={u∈P(Ω):∫Ω‍a(x)|u(x)|p(x)dx<∞}
with the norm
(2.8)|u|La(x)p(x)(Ω)=‖u‖p,a=inf⁡{λ>0:∫Ω‍a(x)|u(x)λ|p(x)dx≤1},
then La(x)p(x)(Ω) is a Banach space.Theorem 2.8.
Suppose thatp(x) satisfies (2.4). Let ρ(u)=∫Ω‍a(x)|u(x)|p(x)dx. If u,uk∈La(x)p(x)(Ω), then(1)
Foru≠0,∥u∥p,a=λ if and only if ρ(u/λ)=1.(2)
∥u∥p,a<1(=1;>1) if and only if ρ(u)<1(=1;>1).(3)
If∥u∥p,a>1, then ∥u∥p,ap-≤ρ(u)≤∥u∥p,ap+.(4)
If∥u∥p,a<1, then ∥u∥p,ap+≤ρ(u)≤∥u∥p,ap-.(5)
limk→∞∥uk∥p,a=0 if and only if limk→∞ρ(uk)=0.(6)
∥uk∥p,a→∞ if and only if ρ(uk)→∞.Theorem 2.9.
LetΩ⊂ℝn be a measurable subset. Suppose that g:Ω×ℝ→ℝ is a Caracheodory function and satisfies
(2.9)|g(x,u)|≤α(x)+β|u|(p1(x))/(p2(x))foranyx∈Ω,t∈R,
where pi(x)≥1, i=1,2, α(x)∈Lp2(x)(Ω), α(x)≥0, β≥0 is a constant, then the Nemytsky operator from Lp1(x)(Ω) to Lp2(x)(Ω) defined by (Ngu)(x)=g(x,u(x)) is a continuous and bounded operator.Theorem 2.10.
Assume that0∈Ω¯ and the boundary of Ω possesses the cone property. Suppose that p(x),s(x),q(x)∈C(Ω¯), 0≤s(x)<N for x∈Ω¯. If q(x) satisfies 1≤q(x)<ps*(x) for x∈Ω¯, there is a compact embedding W1,p(x)(Ω)↪L|x|-s(x)q(x)(Ω).Theorem 2.11.
Assume that0∈Ω¯ and the boundary of Ω possesses the cone property. Suppose that p(x),s(x),q(x)∈C(Ω¯), 0≤s(x)≪p(x) for x∈Ω¯. There is a continuous embedding W1,p(x)(Ω)↪L|x|-s(x)ps*(x)(Ω).Proof.
Letu∈W1,p(x)(Ω). Note that
(2.10)∫Ω|u|ps*(x)|x|s(x)dx=∫Ω‍|u|s(x)|u|ps*(x)-s(x)|x|s(x)dx≤C1‖|ux|s(x)‖p/s‖|u|N(p(x)-s(x))/(N-p(x))‖p/(p-s).
By Theorems 2.7 and 2.10, we have ∥u∥p,|x|-p≤C2∥u∥1,p<∞ and ∥u∥p*≤C3∥u∥1,p<∞. So we get
(2.11)∫Ω‍(|ux|s(x))p(x)/s(x)dx=∫Ω‍|ux|p(x)dx<∞,∫Ω‍|u|(N(p(x)-s(x))/(N-p(x)))⋅(p(x)/(p(x)-s(x)))dx=∫Ω‍|u|p*(x)dx<∞.
Furthermore, we obtain ∫Ω‍|u|ps*(x)/|x|s(x)dx<∞. This shows W1,p(x)(Ω)⊂L|x|-s(x)ps*(x)(Ω), then by the closed graph theorem in Banach space, we get the continuous embedding W1,p(x)(Ω)↪L|x|-s(x)ps*(x)(Ω).


## 3. The Principle of Concentration Compactness

In this section, we will establish the principle of concentration compactness inW01,p(x)(Ω).We denote byℳ(Ω¯) the space of finite nonnegative Borel measures on Ω¯. A sequence μn→μ weakly-* in ℳ(Ω¯) is defined by (μn,u)→(μ,u), for any u∈C(Ω¯)⋂‍C∞(Ω).We first give two lemmas. From [13] we can obtain the proof of the following lemmas. Assume that  p(x) is Lipschitz continuous satisfying (2.4) and s(x) is continuous on Ω¯.Lemma 3.1.
Let{un}⊂L|x|-s(x)p(x)(Ω) be bounded, and un→u∈L|x|-s(x)p(x)(Ω) a.e. on Ω, then
(3.1)lim⁡n→∞∫Ω‍|un|p(x)|x|s(x)-|un-u|p(x)|x|s(x)dx=∫Ω‍|u|p(x)|x|s(x)dx.Lemma 3.2.
Letδ>0, 0<r<R<1, and r/R≤k(δ)=min{exp(-(δ/(2C~))n/p-(1-n)),e-|sn-1|1/(n-1)}, where C~=((1/((1+(δ/2))1/(p+-1)-1))+1)p+-1max{2Cp+,2Cp-}|sn-1|p-/n, |sn-1| denotes the surface area of the unit sphere in ℝn and C satisfies the inequality ∥u∥p*(x)≤C∥∇u∥p(x). Then for every u∈W01,p(x)(Ω),
(3.2)∫Br(x0)‍|u|ps*(x)|x|s(x)dx≤C*max⁡{(∫BR(x0)‍|∇u|p(x)+|u|p(x)dx+δmax⁡{‖u‖1,pp+,‖u‖1,pp-})ps*-/p+,≤C*max⁡≤(∫BR(x0)‍|∇u|p(x)+|u|p(x)dx+δmax⁡{‖u‖1,pp+,‖u‖1,pp-})ps*+/p-},
where C*=sup{∫Ω‍|u|ps*(x)/|x|s(x)dx:∥u∥1,p≤1,u∈W01,p(x)(Ω)}.Theorem 3.3.
Let{un}⊂W01,p(x)(Ω) with ∥un∥1,p≤1 such that
(3.3)un⇀uweaklyinW01,p(x)(Ω),|∇un|p(x)+|un|p(x)⟶μweakly-*inM(Ω¯),|un|ps*(x)|x|s(x)⟶νweakly-*inM(Ω¯),
as n→∞. Then the limit measures are of the form
(3.4)μ=|∇u|p(x)+|u|p(x)+∑j∈J‍μjδxj+μ0δ0+μ~,μ(Ω¯)≤1,ν=|u|ps*(x)|x|s(x)+∑j∈J‍νjδxj+ν0δ0,ν(Ω¯)≤C*,
where J is a countable set, {μj}⊂[0,∞), {νj}⊂[0,∞), μ0≥0, ν0≥0, {xj}∈Ω¯, μ~∈ℳ(Ω¯) is a nonatomic positive measure. δxj and δ0 are atomic measures which concentrate on xj and 0, respectively. C* is as defined in Lemma 3.2. The atoms and the regular part satisfy the generalized Sobolev inequalities
(3.5)ν(Ω¯)≤C*max⁡⁡{μ(Ω¯)ps*+/p-,μ(Ω¯)ps*-/p+},νj≤C*max⁡⁡{μjps*+/p-,μjps*-/p+},ν0≤C*max⁡⁡{μ0ps*+/p-,μ0ps*-/p+}.Proof.
By Lemma3.2, for every δ>0, there exists k(δ)>0 such that for 0<r<R with r/R≤k(δ),
(3.6)∫Br(0)‍|un|ps*(x)|x|s(x)dx≤C*max⁡{(∫BR(0)‍|∇un|p(x)+|un|p(x)dx+δmax⁡{‖un‖1,pp+,‖un‖1,pp-})ps*-/p+≤C*max⁡≤(∫BR(0)‍|∇un|p(x)+|un|p(x)dx+δmax⁡{‖un‖1,pp+,‖un‖1,pp-})ps*+/p-}.
Let η1∈C0∞(Br(0)) and η2∈C0∞(B2R(0)) such that 0≤η1,η2≤1,  η1≡1 in Br/2(0) and η2≡1 in BR(0). Then we have
(3.7)∫Br(0)‍|un|ps*(x)|x|s(x)η1dx⟶∫Br(0)‍η1dν,∫B2R(0)‍(|∇un|p(x)+|un|p(x))η2dx⟶∫B2R(0)‍η2dμ.
Thus,
(3.8)∫Br(0)‍η1dν≤C*max⁡{(∫B2R(0)‍η2dμ+δ)ps*-/p+,(∫B2R(0)‍η2dμ+δ)ps*+/p-}.
Furthermore,
(3.9)ν({0})≤ν(Br/2(0))≤C*max⁡{(μ(B2R(0))+δ)ps*-/p+,(μ(B2R(0))+δ)ps*+/p-}.
Let δ→0 and R→0, then we get
(3.10)ν({0})≤C*max⁡{μ({0})ps*-/p+,μ({0})ps*+/p-},
that is,
(3.11)ν0≤C*max⁡{μ0ps*-/p+,μ0ps*+/p-}.
By Theorem 2.11 and the definition of C*, we have
(3.12)∫Ω‍|u|ps*(x)|x|s(x)dx≤C*max⁡{(∫Ω‍|∇u|p(x)+|u|p(x)dx)ps*-/p+,(∫Ω‍|∇u|p(x)+|u|p(x)dx)ps*+/p-}.
Similar to the proof of Theorem 3.1 in [13], we get
(3.13)ν=|u|ps*(x)|x|s(x)+∑j∈J‍νjδxj+ν0δ0
and the other results.


## 4. Existence of Solutions

LetO(N) be the group of orthogonal linear transformations in ℝN, and G is a subgroup of O(N). For x≠0, we denote the cardinality of Gx={gx:g∈G} by |Gx| and set |G|=infx∈ℝN,x≠0|Gx|. An open subset Ω⊂ℝN is G-invariant if gΩ=Ω for any g∈G.Definition 4.1.
LetΩ be a G-invariant open subset of ℝN. The action of G on W01,p(x)(Ω) is defined by gu(x)=u(g-1x) for any u∈W01,p(x)(Ω). The subspace of invariant functions is defined by
(4.1)W0,G1,p(x)(Ω)={u∈W01,p(x)(Ω):gu=u,∀g∈G}.A functionalI:W01,p(x)(Ω)→ℝN is G-invariant if I∘g=I for any g∈G.
Set(4.2)I(u)=∫Ω‍1p(x)(|∇u|p(x)+|u|p(x))-h(x)ps*(x)|u|ps*(x)|x|s(x)-F(x,u)dx,F(x,t)=∫0t‍f(x,s)ds.
The critical points of I(u), that is,
(4.3)0=I′(u)φ=∫Ω‍|∇u|p(x)-2∇u∇φ+|u|p(x)-2uφ-h(x)|u|ps*(x)-2u|x|s(x)φ-f(x,u)φdx
for all φ∈W01,p(x)(Ω), are weak solutions of the problem (1.1). So next we need only to consider the existence of nontrivial critical points of I(u).
In this paper, assume thatG=O(N) and Ω is O(N)-invariant. By (F-3) and (F-5), we get that I is O(N)-invariant. By the principle of symmetric criticality of Krawcewicz and Marzantowicz [20], u is a critical point of I if and only if u is a critical point of I~=I|W0,O(N)1,p(x)(Ω). So we only need to prove the existence of critical points of I~ on W0,O(N)1,p(x)(Ω).Lemma 4.2.
Any(PS)c sequence {un}⊂W0,O(N)1,p(x)(Ω)possesses a convergent subsequence.Proof.
Suppose thatI~(un)→c, c∈ℝ, and I~′(un)→0 in (W0,O(N)1,p(x)(Ω))*. Let l(x)=(p(x)+ps*(x))/2 and |∇(1/l(x))|≤C. Denote a=infx∈Ω¯((1/p(x))-(1/l(x)))>0 and b=infx∈Ω¯((1/(l(x)))-(1/(ps*(x))))>0. Then we have
(4.4)I~(un)-〈I~′(un),unl(x)〉=∫Ω‍(1p(x)-1l(x))(|∇un|p(x)+|un|p(x))+h(x)(1l(x)-1ps*(x))|un|ps*(x)|x|s(x)+1l(x)f(x,un)un-F(x,un)dx-∫Ω‍|∇un|p(x)-2∇un∇(1l(x))undx≥∫Ω‍a(|∇un|p(x)+|un|p(x))+bh(x)|un|ps*(x)|x|s(x)+1l(x)f(x,un)un-F(x,un)dx-∫Ω‍|∇un|p(x)-2∇un∇(1l(x))undx.
By Young's inequality, for ɛ1∈(0,1), we get
(4.5)||∇un|p(x)-2∇un⋅un|≤ɛ1|∇un|p(x)+ɛ1|un|ps*(x)+C(ɛ1).
By (F-2), |(1/l(x))f(x,un)un-F(x,un)|≤C(|un|+|un|q(x)), then we have for ɛ2∈(0,1)(4.6)|un|+|un|q(x)≤ɛ2|un|ps*(x)+C(ɛ2).
From h(x)/|x|s(x)→∞ as x→0, we get that there exists H¯>0 such that h(x)/|x|s(x)>H¯ for any x∈Ω, so we have
(4.7)I~(un)-〈I~′(un),unl(x)〉≥∫Ω‍a(|∇un|p(x)+|un|p(x))dx+∫Ω‍bH¯|un|ps*(x)dx-Cɛ1∫Ω‍|∇un|p(x)dx-C(ɛ1+ɛ2)∫Ω‍|un|ps*(x)dx-C(ɛ1)-C(ɛ2).
Take ɛ1 and ɛ2 sufficiently small such that Cɛ1<a/2 and C(ɛ1+ɛ2)≤bH¯, thus,
(4.8)c+1>I(un)≥∫Ω‍a2(|∇un|p(x)+|un|p(x))dx-C,
if n is sufficiently large. Furthermore, we obtain ∥un∥1,p<∞.
Note that(4.9)∫Ω‍(|∇un|p(x)-2∇un-|∇u|p(x)-2∇u)(∇un-∇u)dx≤≤|〈I~′(un),un-u〉|+∫Ω‍|(|un|p(x)-2un-|u|p(x)-2u)(un-u)|dx≤≤+|〈I~′(u),un-u〉|+∫Ω‍|h(x)(|un|ps*(x)-2un|x|s(x)-|u|ps*(x)-2u|x|s(x))(un-u)|dx≤≤+∫Ω‍|(f(x,un)-f(x,u))(un-u)|dx≤≜∑i=15‍Ii.
Because {un} is bounded in W0,O(N)1,p(x)(Ω), there exists a subsequence (still denoted by un) such that un⇀u weakly in W0,O(N)1,p(x)(Ω). Then we have un→u in Lq(x)(Ω). It is easy to get I1→0, I2→0, and I3→0. By (F-2)
(4.10)∫Ω‍|f(x,un)|q′(x)dx≤∫Ω‍(c1+c2|un|q(x)-1)q′(x)dx≤C∫Ω‍(1+|un|)(q(x)-1)q′(x)dx≤C(|Ω|+∫Ω‍|un|q(x)dx).
Then we have that ∥f(x,un)∥q′ is bounded. By
(4.11)I5≤2‖f(x,un)‖q′‖un-u‖q+2‖f(x,u)‖q′‖un-u‖q,
we get I5→0.
Next we show thatI4→0. Note that
(4.12)I4≤h0(∫Ω‍|un|ps*(x)-1|x|s(x)|un-u|dx+∫Ω‍|u|ps*(x)-1|x|s(x)|un-u|dx)≤2h0(‖|un|ps*(x)-1|x|s(x)/ps*′(x)‖ps*′‖un-u|x|s(x)/ps*(x)‖ps*+‖|u|ps*(x)-1|x|s(x)/ps*′(x)‖ps*′‖un-u|x|s(x)/ps*(x)‖ps*),
where h0=maxx∈Ω¯h(x). By Theorem 2.11, ∥|un|ps*(x)-1/|x|s(x)/ps*′(x)∥ps*′ is bounded. If we show that there exists a subsequence (still denoted by {un}) such that ∫Ω‍|un-u|ps*(x)/|x|s(x)dx→0 as n→∞, then I4→0.
Asun⇀u weakly in W0,O(N)1,p(x)(Ω), passing to a subsequence, still denoted by {un}, by Theorem 3.3 we assume that there exist μ,ν∈ℳ(Ω¯) and {xj}j∈J in Ω¯ such that |∇un|p(x)+|un|p(x)→μ weakly-* in ℳ(Ω¯) and |un|ps*(x)/|x|s(x)→ν weakly-* in ℳ(Ω¯), where
(4.13)μ=|∇u|p(x)+|u|p(x)+∑j∈J‍μjδxj+μ0δ0+μ~,ν=|u|ps*(x)|x|s(x)+∑j∈J‍νjδxj+ν0δ0.J is a countable set, {μj}⊂[0,∞), {νj}⊂[0,∞), μ0≥0, ν0≥0, μ~∈ℳ(Ω¯) is a nonatomic positive measure. Take η≡1, then
(4.14)lim⁡n→∞∫Ω‍|un|ps*(x)|x|s(x)ηdx=∫Ω‍ηdν=∫Ω‍|u|ps*(x)|x|s(x)dx+∑j∈J‍νj+ν0.
We claim ν0=0 and νj=0 for any j∈J. First we consider ν0.
For anyɛ>0, choose φ0∈C0∞(B2ɛ(0)) such that 0≤φ0≤1, φ0=1 on Bɛ(0) and |∇φ0|≤2/ɛ. Then
(4.15)〈I~′(un),unφ0〉=∫Ω‍(|∇un|p(x)+|un|p(x))φ0dx-∫Ω‍h(x)|un|ps*(x)φ0|x|s(x)dx-∫Ω‍f(x,un)unφ0dx+∫Ω‍|∇un|p(x)-2∇un∇φ0undx.
We have
(4.16)lim⁡n→∞∫B2ɛ(0)‍(|∇un|p(x)+|un|p(x))φ0dx=∫B2ɛ(0)‍φ0dμ,lim⁡n→∞∫B2ɛ(0)‍|un|ps*(x)φ0|x|s(x)dx=∫B2ɛ(0)‍φ0dν.
By Theorem 2.1,
(4.17)∫B2ɛ(0)‍||∇un|p(x)-2∇un∇φ0un|dx≤2‖un∇φ0‖p,B2ɛ(0)‖|∇un|p(x)-1‖p′,B2ɛ(0)≤C‖un∇φ0‖p,B2ɛ(0).
By Theorem 2.6, we have un→u in Lp(x)(Ω), then
(4.18)lim⁡n→∞∫B2ɛ(0)‍|un∇φ0|p(x)dx=∫B2ɛ(0)‍|u∇φ0|p(x)dx.
Furthermore,
(4.19)∫B2ɛ(0)‍|u∇φ0|p(x)dx≤2‖|∇φ0|p(x)‖N/p,B2ɛ(0)‖|u|p(x)‖N/(N-p),B2ɛ(0),∫B2ɛ(0)‍|∇φ0|Ndx≤4NωN,
where ωN is the volume of the unit boll. By limɛ→0∫B2ɛ(0)‍|u|p*(x)dx=0, then we have
(4.20)lim⁡ɛ→0lim⁡n→∞∫Ω‍||∇un|p(x)-2∇un∇φ0un|dx=0.
Since ∥f(x,un)∥q′ is bounded and by Theorem 2.9 we have
(4.21)lim⁡n→∞∫B2ɛ(0)‍|f(x,un)-f(x,u)|q′(x)dx=0.
From
(4.22)∫B2ɛ(0)‍|f(x,un)un-f(x,u)u|dx≤2‖f(x,un)‖q′‖un-u‖q+2‖f(x,un)-f(x,u)‖q′‖u‖q,
we have
(4.23)lim⁡n→∞∫B2ɛ(0)‍f(x,un)unφ0dx=∫B2ɛ(0)‍f(x,u)uφ0dx.
Therefore,
(4.24)lim⁡ɛ→0lim⁡n→∞∫B2ɛ(0)‍f(x,un)unφ0dx=lim⁡ɛ→0∫B2ɛ(0)‍f(x,u)uφ0dx=0.
Thus, we have
(4.25)0=lim⁡n→∞〈I~′(un),unφ0〉=∫B2ɛ(0)‍φ0dμ-∫B2ɛ(0)‍h(x)φ0dν-∫B2ɛ(0)‍f(x,u)uφ0dx+lim⁡n→∞∫B2ɛ(0)‍|∇un|p(x)-2∇un∇φ0⋅undx.
Furthermore, we obtain
(4.26)0=lim⁡ɛ→0lim⁡n→∞〈I~′(un),unφ0〉=μ0-h(0)ν0.
As h(0)=0, μ0=0, thus, ν0=0.
Next we considerνj for any j∈J. Suppose ∃j0∈J such that νj0>0. Note that un∈W0,O(N)1,p(x)(Ω), then for any g∈O(N), ν(gxj0)=ν(xj0)>0. By |O(N)|=∞, we get ν({gxj0:g∈O(N)})=∞. As the measure ν is finite, that is a contradiction. So we obtain that ν0=0 and νj=0 for any j∈J. Thus,
(4.27)lim⁡n→∞∫Ω‍|un|ps*(x)|x|s(x)dx=∫Ω‍|u|ps*(x)|x|s(x)dx.
By Lemma 3.1, we obtain limn→∞∫Ω‍|un-u|ps*(x)/|x|s(x)dx=0, that is, un→u strongly in L|x|-s(x)ps*(x)(Ω).
We obtain that{un} possesses a subsequence (still denoted by {un}), such that Ii→0, i=1,…,5, as n→∞. Thus, ∫Ω‍(|∇un|p(x)-2∇un-|∇u|p(x)-2∇u)(∇un-∇u)dx→0, as n→∞. As in the proof of Theorem 3.1 in [5], we divide Ω into two parts:
(4.28)Ω1={x∈Ω:p(x)≥2},Ω2={x∈Ω:p(x)<2}.
We have
(4.29)∫Ω1‍|∇un-∇u|p(x)dx+∫Ω2‍|∇un-∇u|p(x)dx⟶0,
that is, ∫Ω‍|∇un-∇u|p(x)dx→0. Then un→u in W0,O(N)1,p(x)(Ω).SinceW01,p(x)(Ω) is a separable and reflexive Banach space, W0,O(N)1,p(x)(Ω) is also a separable and reflexive Banach space. So there exist {en}n=1∞⊂W0,O(N)1,p(x)(Ω) and {en*}n=1∞⊂(W0,O(N)1,p(x)(Ω))* such that
(4.30)〈ej*,ei〉={1,i=j,0,i≠j,W0,O(N)1,p(x)(Ω)=span⁡{en:n=1,2,…}¯,(W0,O(N)1,p(x)(Ω))*=span⁡{en*:n=1,2,…}¯.
For k=1,2,…, denote Xk=span{ek}, Yk=⊕j=1kXj, Zk=⊕j=k∞Xj¯.Theorem 4.3.
Under assumptions (F-1)–(F-5), the problem (1.1)  admits a sequence of solutions{un}⊂W0,O(N)1,p(x)(Ω) such that I(un)→∞.Proof.
Setφ(u)=∫Ω‍F(x,u)dx. We first show that φ(u) is weakly strongly continuous. Let un⇀u weakly in W0,O(N)1,p(x)(Ω). So we have un→u in Lq(x)(Ω). Note that
(4.31)|F(x,u)|≤C(|u|+|u|q(x))≤C(1+|u|q(x)),
then by Theorem 2.9 we obtain F(x,un)→F(x,u) in L1(Ω). By Proposition 3.5 in [18],
(4.32)βk=βk(r)=sup⁡u∈Zk,‖u‖1,p≤r∫Ω‍|F(x,u)|dx⟶0,
as k→∞ for r>0.
Set(4.33)θk=θk(r)=sup⁡u∈Zk,‖u‖1,p≤r∫Ω‍|u|ps*(x)|x|s(x)dx.
Next we show θk→∑j∈Jνj+ν0 as k→∞. Note that 0≤θk+1≤θk, then θk→θ≥0, as k→∞. There exists uk∈Zk with ∥uk∥1,p≤r such that 0≤θk-∫Ω‍(|uk|ps*(x)/|x|s(x))dx<1/k, for each k=1,2,…. As W0,O(N)1,p(x)(Ω) is reflexive, passing to a subsequence, still denoted by {uk}, we assume uk⇀u weakly in W0,O(N)1,p(x)(Ω). We claim u=0. In fact, for any em*, we have em*(uk)=0, when k>m, then em*(uk)→0 as k→∞. It is immediate to get em*(u)=0 for any m∈ℕ. Then we have u=0. By Theorem 3.3, there exist a finite measure ν and a sequence {xj}⊂Ω¯ such that
(4.34)|uk|ps*(x)|x|s(x)⇀ν=|u|ps*(x)|x|s(x)+∑j∈J‍νjδxj+ν0δ0,
where J is countable. Set η≡1, we obtain ∫Ω‍(|uk|ps*(x)/|x|s(x))ηdx→∑j∈Jνj+ν0. So we have limk→∞θk=∑j∈Jνj+ν0≤ν(Ω¯)<∞.
For anyn∈ℕ, there exists a positive integer kn such that βk(n)≤1 and θk(n)≤∑j∈Jνj+ν0+1 for all k≥kn. Assume that kn<kn+1 for each n. Define {rk:k=1,2,…} in the following way:
(4.35)rk={n,kn≤k<kn+1,1,1≤k<k1.
Then we get rk→∞ as k→∞. Hence, for u∈Zk with ∥u∥1,p=rk, we get
(4.36)I~(u)≥1p+‖u‖1,pp--h0ps*-θk(rk)-βk(rk)≥1p+‖u‖1,pp--h0ps*-(∑j∈Jvj+v0+1)-1,
where h0 is as defined in Lemma 4.2. So
(4.37)inf⁡u∈Zk,∥u∥1,p=rkI~(u)⟶∞ask⟶∞.
Note that forɛ∈(0,1),   |F(x,u)|≤Cɛ|u|ps*(x)+C(ɛ), then
(4.38)∫Ω‍F(x,u)dx≤Cɛ∫Ω‍|u|ps*(x)dx+C(ɛ)|Ω|.
We have
(4.39)I~(u)≤∫Ω‍|∇u|p(x)+|u|p(x)p(x)dx-∫Ω‍H¯|u|ps*(x)ps*+dx+Cɛ∫Ω‍|u|ps*(x)dx+C(ɛ)|Ω|.
Take ɛ sufficiently small so that Cɛ≤H¯/2ps*+, then
(4.40)I~(u)≤∫Ω‍|∇u|p(x)+|u|p(x)dx-m∫Ω‍|u|ps*(x)dx+C,
where m=H¯/2ps*+. Since the dimension of Yk is finite, any two norms on Yk are equivalent, then k1∥u∥1,p≤∥u∥ps*≤k2∥u∥1,p, k1,k2>0. As in the proof of Theorem 4.2 in [13], we can find hypercubes {Qi}i=1Q which mutually have no common points such that Ω¯⊆⋃i=1Q‍Qi¯ and pi+=supy∈Ωip(y)<infy∈Ωips*(y)=psi*-, where Ωi=Qi⋂‍Ω. Then we have
(4.41)I~(u)≤∑‖u‖1,p,Ωi>1‍(‖u‖1,p,Ωipi+-mk2psi*-‖u‖1,p,Ωipsi*-)+∑‖u‖1,p,Ωi≤1‍(‖u‖1,p,Ωipi--mk2psi*+‖u‖1,p,Ωipsi*+)+C≤∑‖u‖1,p,Ωi>1‍(‖u‖1,p,Ωipi+-mk2psi*-‖u‖1,p,Ωipsi*-)+Q+C.
Let fi(t)=tpi+-mk2psi*-tpsi*-, for i=1,…,Q. Take si>0 such that fi(si)=maxt≥0fi(t)≥fi(0)=0. Denote gi(t)=tpi+-mk2psi*-tpsi*-+∑j=1Q‍fj(sj)+Q+C, for i=1,…,Q. By limt→∞gi(t)=-∞, there exists t0>0 such that gi(t)≤0 for t∈[t0,+∞), foralli=1,…,Q. For any k=1,2,…, take ∥u∥1,p=ρk=max{Qt0,rk+1}. Note that ∃i0 such that
(4.42)‖u‖1,p,Ωi0≥1Q∑i=1Q‍‖u‖1,p,Ωi≥ρkQ≥t0.
Then we have gi0(∥u∥1,p,Ωi0)≤0. Thus,
(4.43)I~(u)≤gi0(‖u‖1,p,Ωi0)=∑i=1Q‍fi(si)+fi0(‖u‖1,p,Ωi0)+Q+C≤0.
Therefore, I~(u)≤0 for u∈YK⋂‍Sρk, where Sρk={u:∥u∥1,p=ρk}. From Lemma 4.2 we have that I~(u) satisfies (PS)c condition. In view of (F-4), by Fountain Theorem [21], we conclude the result.



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*Source: 894925-2012-08-26.xml*