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INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH†
INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH†
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 137-152
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : April 03, 2013
  • Accepted : June 26, 2013
  • Published : January 28, 2014
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CHENXING ZHOU
SIHUA LIANG

Abstract
In this paper, we prove, in the spirit of [3 , 12 , 20 , 22 , 23] , the existence of infinitely many small solutions to the following quasilinear elliptic equation −Δ p(x) u +| u | p(x)−2 u =| u | q(x)−2 u + λƒ ( x, u ) in a smooth bounded domain Ω or ℝ N . We also assume that where p ( x ) = Np ( x )/( N - p ( x )) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22] , and property of these solutions are also obtained. AMS Mathematics Subject Classification : 35J60, 35B33.
Keywords
1. Introduction
In this paper we deal with quasilinear elliptic problem of the form
PPT Slide
Lager Image
where Ω ⊂ ℝ N ( N ≥ 3) is a bounded domain with smooth boundary and p ( x ), q ( x ) are two continuous functions on
PPT Slide
Lager Image
where denote by p ( x ) ≪ q ( x ) the fact that inf x∈Ω ( q ( x ) − p ( x )) > 0. λ is a positive parameter, Δ p(x) u := div(|∇ u | p(x)−2 u ) is the p ( x )-Laplacia operator. On the exponent q ( x ) we assume that is the critical exponent in the sense that
PPT Slide
Lager Image
is the critical exponent according to the Sobolev embedding. Our goal will be to obtain infinitely many small weak solutions which tend to zero for (1) in the generalized Sobolev space
PPT Slide
Lager Image
for the general nonlinearities of the type ƒ ( x, u ).
The study of differential equations and variational problems involving variable exponent conditions has been a very interesting and important topic. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, image processing and so on. For example, Chen, Levin and Rao [4] proposed the following model in image processing
PPT Slide
Lager Image
where p ( x ) is a function satisfies 1 ≤ p ( x ) ≤ 2 and ƒ is a convex function. For more information on modelling physical phenomena by equations involving p ( x )-growth condition we refer to [1 , 19 , 28 , 30] . The appearance of such physical models was facilitated by the development of variable Lebesgue and Sobolev spaces, L p(x) and W 1,p(x) , where p ( x ) is a real-valued function. On the variable exponent Sobolev spaces which have been used to study p ( x )-Laplacian problems, we refer to [5 , 21 , 29] . On the existence of solutions for elliptic equations with variable exponent, we refer to [2 , 6 , 7 , 8 , 9 , 10 , 11 , 16 , 17 , 31] .
In recent years, the existence of infinitely many solutions have been obtained by many papers. When p ( x ) ≡ p = 2 (a constant) with Dirichlet boundary condition, Li and Zou [23] studied a class of elliptic problems with critical exponents, they obtained the existence theorem of infinitely many solutions under suitable hypotheses. He and Zou [20] proved that the existence infinitely many solutions under case the general nonlinearities. When p ( x ) ≡ p ≠ 2. Ghoussoub and Yuan [18] obtained the existence of infinitely many nontrivial solutions for Hardy-Sobolev subcritical case and Hardy critical case by establishing Palais-Smale type conditions around appropriate chosen dual sets in bounded domain. Li and Zhang [24] studied the existence of multiple solutions for the nonlinear elliptic problems of p&q -Laplacian type involving the critical Sobolev exponent, they obtained infinitely many weak solutions by using Lusternik-Schnirelman’s theory for Z 2 -invariant functional.
On the existence of infinitely many solutions for p ( x )-Laplacian problems have been studied by [2 , 7 , 9 , 31] , but they did not give any further information on the sequence of solutions. Moreover, these papers deal with subcritical nonlinearities. Very little is known about critical growth nonlinearities for variable exponent problems [14 , 15] , since one of the main techniques used in order to deal with such issues is the concentration-compactness principle. This result was recently obtained for the variable exponent case independently in [12 , 13] . In both of these papers the proof are similar and both relates to that of the original proof of P.L. Lions [25 , 26] .
Recently, Kajikiya [22] established a critical point theorem related to the symmetric mountain pass lemma and applied to a sublinear elliptic equation. But there are no such results on p ( x )-Laplacian problem with critical growth (1).
Motivated by reasons above, the aim of this paper is to show that the existence of infinitely many solutions of problem (1), and there exists a sequence of infinitely many arbitrarily small solutions converging to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya [22] . In order to use the symmetric mountain-pass lemma, there are many difficulties. The main one in solving the problem is a lack of compactness which can be illustrated by the fact that the embedding of W 1,p(x) (Ω) into L p*(x) (Ω) is no longer compact. Hence the concentration-compactness principle is used here to overcome the difficulty. The main result of this paper is as follows.
Theorem 1.1. Suppose that f ( x, u ) satisfies the following conditions:
  • (H1)ƒ(x, u) ∈C(Ω×R, R),ƒ(x, -u) =-ƒ(x, u)for all u∈R;
  • (H2)
  • (H3)
Then there exists λ such that for any λ ∈ (0, λ ), problem (1) has a sequence of non-trivial solutions { un } and un → 0 as n → ∞.
Remark 1.1. If without the symmetry condition (i.e. ƒ ( x,−u ) = ( x, u )) in Theorem 1.1, we get an existence theorem of at least one nontrivial solution to problem (1) by the same method in this paper.
Remark 1.2. In this paper, we use concentration-compactness principle due to [12] which is slightly more general than those in [13] , since we do not require q ( x ) to be critical everywhere.
Remark 1.3. There exist many functions ƒ ( x, t ) satisfy condition ( H 1 )-( H 3 ), for example, ƒ ( x, u ) = u (p--1)/3 , where p > 1.
Remark 1.4. Theorem 1.1 is new as far as we know and it generalizes results in [3] for p ( x )-Laplacian type problem. We mainly follow the way in [3] to prove our main result.
Definition 1.2. We say that
PPT Slide
Lager Image
is a weak solution of problem (1) if for any
PPT Slide
Lager Image
PPT Slide
Lager Image
The energy functional corresponding to problem (1) is defined as follows,
PPT Slide
Lager Image
then, it is easy to check that as arguments [27] show that J ( u ) is well defined on
PPT Slide
Lager Image
and the weak solutions for problem (1) coincides with the critical points of J . We try to use a new version of the symmetric mountain-pass lemma due to Kajikiya [22] . But since the functional J ( u ) is not bounded from below, we could not use the theory directly. So we follow [3] to consider a truncated functional of J ( u ). Denote J ′ : E E is the derivative operator of J in the weak sense. Then
PPT Slide
Lager Image
Definition 1.3. We say J satisfies Palais-Smale condition (( PS ) for short) in
PPT Slide
Lager Image
which satisfies that { J ( un )} is bounded and ∥ J ′( un )∥ p(x) → 0 as n → ∞, has a convergent subsequence.
Under assumption ( H 2 ), we have
PPT Slide
Lager Image
which means that, for all ε > 0, there exist a ( ε ), b ( ε ) > 0 such that
PPT Slide
Lager Image
PPT Slide
Lager Image
Hence, for any constants β we have
PPT Slide
Lager Image
for some c ( ε ) > 0.
The remainder of the paper is organized as follows. In Section 2, we shall present some basic properties of the variable exponent Sobolev spaces. In Section 3, we will prove the corresponding energy functional satisfies the ( PS ) condition. In Section 4, we shall prove our main results.
2. Weighted variable exponent Lebesgue and Sobolev spaces
We recall some definitions and properties of the variable exponent Lebesgue-Sobolev spaces L p(·) (Ω) and W 1,p(·) (Ω), where Ω is a bounded domain in ℝ N .
Set
PPT Slide
Lager Image
For any
PPT Slide
Lager Image
we define
PPT Slide
Lager Image
We can introduce the variable exponent Lebesgue space as follows:
L p(·) (Ω) = { u : u is a measurable real-valued function such that
PPT Slide
Lager Image
for
PPT Slide
Lager Image
Equipping with the norm on L p(x) (Ω) by
PPT Slide
Lager Image
which is a Banach space, we call it a generalized Lebesgue space.
Proposition 2.1 ( [5 , 11] ). (i) The space ( L p(x) (Ω), | · | p(x) ) is a separable, uni-form convex Banach space, and its conjugate space is L q(x) (Ω), where 1/ q ( x ) + 1/ p ( x ) = 1. For any u L p(x) (Ω) and v L q(x) (Ω), we have
PPT Slide
Lager Image
(ii) If 0 < |Ω| < ∞ and p 1 , p 2 are variable exponents in
PPT Slide
Lager Image
such that p 1 p 2 in Ω, then the embedding L p2(·) (Ω) ,→ L p1(·) (Ω) is continuous .
Proposition 2.2 ( [5 , 11] ). The mapping ρ p(·) : L p(·) (Ω) → ℝ defined by
PPT Slide
Lager Image
Then the following relations hold:
PPT Slide
Lager Image
Next, we define W 1,p(x) (Ω) is defined by
PPT Slide
Lager Image
and it can be equipped with the norm
PPT Slide
Lager Image
Denote
PPT Slide
Lager Image
under the norm
PPT Slide
Lager Image
We know that if Ω ⊂ ℝ N is a bounded domain, ∥ u ∥ and ∥ u 1 are equivalent norms on
PPT Slide
Lager Image
Proposition 2.3 ( [5 , 11] ). (i) W 1,p(x) (Ω) are separable re exive Banach spaces; (ii) If
PPT Slide
Lager Image
then the embedding W 1,p(x) (Ω) → L q(x) (Ω) is continuous .
In this paper, we use the following equivalent norm on W 1,p(x) (Ω):
PPT Slide
Lager Image
Proposition 2.4 ( [21 , 6] ). Let I ( u ) = ƒ Ω |∇ u | p(x) + | u | p(x) dx . If u, un W 1,p(x) (Ω), then the following relations hold:
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
3. Preliminaries and lemmas
In the following, we always use C and ci ( i = 1, 2, · · · ) to denote positive constants. We give the concentration-compactness principle of the variable exponent due to [12 , 15] .
Lemma 3.1. Let q ( x ) and p ( x ) be two continuous functions such that
PPT Slide
Lager Image
Let { uj } j∈ℕ be a weakly convergent sequence in
PPT Slide
Lager Image
with weak limit u, and such that |∇ uj | p(x) μ weakly - in the sense of measures; | uj | q(x) v weakly - in the sense of measures. Assume, moreover that
PPT Slide
Lager Image
Then, for some countable index set I we have
(i) ν = | u | q(x) + Σ i∈I νi δ xi , νi > 0;
(ii) μ ≥ |∇ u | p(x) + Σ i∈I μi δ xi , μi > 0;
(iii)
PPT Slide
Lager Image
where { xi } iI ⊂ Γ and S is the best constant in the Gagliardo-Nirenberg-Sobolev inequality for variable exponents, namely
PPT Slide
Lager Image
In order to prove the functional J satisfies the local ( PS ) c condition, we take continuous function η ( x ) satisfies
PPT Slide
Lager Image
Denote
PPT Slide
Lager Image
PPT Slide
Lager Image
Lemma 3.2. Assume condition ( H 2 ) holds. Then for any λ > 0, there exists positive constant m > 0 such that the functional J satisfies the local ( PS ) c condition in
PPT Slide
Lager Image
in the following sense: if
PPT Slide
Lager Image
and J ′( un ) → 0 for some sequence in
PPT Slide
Lager Image
Then { un } contains a subse-quence converging strongly in
PPT Slide
Lager Image
Proof . First, we show that { un } is bounded in
PPT Slide
Lager Image
If ∥ un p(x) → ∞ as n → ∞. Thus, we may assume that ∥ un p(x) > 1 for any integer n .
Then for n sufficiently large, we have
PPT Slide
Lager Image
By (4), for any ( x, t ) ∈ Ω × ℝ, we have
PPT Slide
Lager Image
On the other hand, noting that p ( x ) ≪ q ( x ), by the Young inequality, for any ε 2 , ε 3 ∈ (0, 1), we get
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Thus, relations (13)-(16) imply that
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Thus, we choose ε 2 , ε 3 be so small that d 1 c 1 ε 2 > 0 and
PPT Slide
Lager Image
It follows from (8) and (17) that { un } is bounded in
PPT Slide
Lager Image
Therefore we can assume that
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Note that if I = ∅ then un u strongly in L q(x) (Ω). If not, let xi be a singular point of the measures μ and ν , define a function
PPT Slide
Lager Image
such that ϕ ( x ) = 1 in B ( xi , ε ), ϕ ( x ) = 0 in Ω \ ( xi , 2 ε ) and |∇ ϕ | ≤ 2/ ε in Ω. As
PPT Slide
Lager Image
we obtain that
PPT Slide
Lager Image
i.e.
PPT Slide
Lager Image
On the other hand, by Hölder inequality and boundedness of { un }, we have that
PPT Slide
Lager Image
From (18), (19) and (21), we get that
PPT Slide
Lager Image
Combing this with Lemma 2.1 (iii), we obtain
PPT Slide
Lager Image
This result implies that
PPT Slide
Lager Image
If the second case νi SN holds, for some i I , then by using Lemma 2.1 and selecting ε 2 , ε 3 in (17) such that
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
This is impossible. Consequently, νi = 0 for all i I and hence
PPT Slide
Lager Image
Since { un } is bounded in
PPT Slide
Lager Image
we deduce that there exists a subsequence, again denoted by { un }, and
PPT Slide
Lager Image
such that { un } converges weakly to
PPT Slide
Lager Image
Note that
PPT Slide
Lager Image
On the other hand, we have
PPT Slide
Lager Image
Using the fact that { un } converges strongly to u 0 in L q(x) (Ω) and inequality (5), we have
PPT Slide
Lager Image
where c 1 c 2 and c 3 are positive constants. Using | un u 0 | q(x) → 0 as n → ∞, we deduce that
PPT Slide
Lager Image
PPT Slide
Lager Image
By (23) and (24), we obtain
PPT Slide
Lager Image
It is known that
PPT Slide
Lager Image
where (· , ·) is the standard scalar product in ℝ N . Relations (25) and (26) yield
PPT Slide
Lager Image
This fact and relation (10) imply ∥ un u 0 p(x) → 0 as n → ∞. The proof is complete.
4. Existence of a sequence of arbitrarily small solutions
In this section, we prove the existence of infinitely many solutions of (1) which tend to zero. Let X be a Banach space and denote
Σ := { A X \ {0} : A is closed in X and symmetric with respect to the orgin}. For A ∈ Σ, we define genus γ ( A ) as
PPT Slide
Lager Image
If there is no mapping φ as above for any m N , then γ ( A ) = +∞. Let Σ k denote the family of closed symmetric subsets A of X such that 0 ∉ A and γ ( A ) ≥ k . We list some properties of the genus (see [22] ).
Proposition 4.1. Let A and B be closed symmetric subsets of X which do not contain the origin. Then the following hold .
  • (1)If there exists an odd continuous mapping from A to B, then γ(A) ≤γ(B);
  • (2)If there exists an odd homeomorphism from A to B, then γ(A) =γ(B);
  • (3)If γ(B) < ∞,then
  • (4)Then n-dimensional sphere Snhas a genus of n+1by the Borsuk-Ulam Theorem;
  • (5)If A is compact, then γ(A) < +∞and there exists δ> 0such that Uδ(A) ∈ Σand γ(Uδ(A)) =γ(A),where Uδ(A) = {x ∈ X : ∥x − A∥ ≤ δ}
The following version of the symmetric mountain-pass lemma is due to Kajikiya [22] .
Lemma 4.2. Let E be an infinite-dimensional space and J C 1 ( E,R ) and suppose the following conditions hold .
  • (C1)J(u)is even, bounded from below, J(0) = 0and J(u)satisfies the Palais-Smale condition;
  • (C2)For each k∈N,there exists an Ak∈ Σksuch thatsupu∈AkJ(u) < 0.Then either(R1)or(R2)below holds.
  • (R1)There exists a sequence{uk}such that J′(uk) = 0,J(uk) < 0and{uk}converges to zero.
  • (R2)There exist two sequences{uk}and{vk}such that J′(uk) = 0,J(uk) < 0,uk≠ 0, limk→∞uk= 0,J′(vk) = 0,J(vk) < 0, limk→∞vk= 0,and{vk}converges to a non-zero limit.
Remark 4.1. From Lemma 4.2 we have a sequence { uk } of critical points such that J ( uk ) ≤ 0, uk ≠ 0 and lim k→∞ uk = 0.
In order to get infinitely many solutions we need some lemmas. We first point out that we have
PPT Slide
Lager Image
Proposition 2.3 (ii) imply that
PPT Slide
Lager Image
where c 4 > 0.
Next, we focus our attention on the case when
PPT Slide
Lager Image
For such a u by relation (9) we obtain
PPT Slide
Lager Image
Using (3) and (27)-(29), we deduce that
PPT Slide
Lager Image
where
PPT Slide
Lager Image
with ∥ u p(x) < 1. If we define
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
From the definition of Q ( s ) and the fact that p + < q + , we konw that there exists λ such that for λ ∈ (0, λ ), Q ( t ) attains its positive maximum, that is, there exists
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
Therefore, for e 0 ∈ (0, e 1 ), we may find R 0 < R 1 such that Q ( R 0 ) = e 0 . Now we define
PPT Slide
Lager Image
Then it is easy to see χ ( t ) ∈ [0, 1] and χ ( t ) is C . Let φ ( u ) = χ (∥ u p(x) ) and consider the perturbation of J ( u ):
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
From the above arguments, we have the following:
Lemma 4.3. Let G ( u ) be defined as in (31). Then
  • (i)G∈C1(E,R)and G is even and bounded from below;
  • (ii)If G(u)
  • (iii)There exist m∗> 0such that SN−m∗> 0,and λ∗such that for λ∈ (0,λ∗),G satisfies a local(PS)ccondition for
Proof . It is easy to see (i) and (ii). (iii) are consequences of (ii) and Lemma 3.2.
Lemma 4.4. Assume that ( H 3 ) of Theorem 1.1 holds. Then for any k N , there exists
PPT Slide
Lager Image
Proof . First, by ( H 3 ) of Theorem 1.1, for any fixed
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
Next, given any k N , let Ek be a k -dimensional subspace of
PPT Slide
Lager Image
We take u Ek with norm ∥ u p(x) = 1, for 0 < ρ < min{ R 0 , 1}, we get
PPT Slide
Lager Image
Since Ek is a space of finite dimension, all the norms in Ek are equivalent. If we define
PPT Slide
Lager Image
It follows from (32)that
PPT Slide
Lager Image
since lim |ρ|→0 M ( ρ ) = +∞. That is,
PPT Slide
Lager Image
This completes the proof.
Now we give the proof of Theorem 1.1.
Proof of Theorem 1.1 Recall that
PPT Slide
Lager Image
and define
PPT Slide
Lager Image
By Lemmas 4.3 (i) and 4.4, we know that −∞ < ck < 0. Therefore, assumptions ( C 1 ) and ( C 2 ) of Lemma 4.2 are satisfied. This means that G has a sequence of solutions { un } converging to zero. Hence, Theorem 1.1 follows by Lemma 4.3 (ii).
BIO
Chenxing Zhou received master’s degree from Jilin University. His research interests boundary value problems and variational problems.
College of Mathematics, Changchun Normal University, Changchun 130032, Jilin, PR China.
e-mail: mathfhmiao@163.com
Sihua Liang received Ph.D at Nanjing Normal University. He is currently a Postdoctor at Jilin University since 2011. His research interests boundary value problems and variational problems.
College of Mathematics, Changchun Normal University, Changchun 130032, Jilin, PR China.
e-mail: liangsihua@126.com
References
Acerbi E. , Mingione G. (2001) Regularity results for a class of functionals with nonstandard growth Arch. Rational Mech. Anal. 156 121 - 140    DOI : 10.1007/s002050100117
Andrei I. (2009) Existence of solutions for a p(x)-Laplacian nonhomogeneous equations E. J. Differential Equations 2009 (72) 1 - 12
Azorero Garcia J , Aloson Peral I. (1991) Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term Trans. Amer. Math. Soc. 323 877 - 895    DOI : 10.2307/2001562
Chen Y. , Levine S. , Rao R. (2006) Functionals with p(x) growth in image processing SIAM J. Appl. Math. 66 1383 - 1406    DOI : 10.1137/050624522
Fan X.L. , Zhao D. (2001) On the spaces Lp(x)(Ω) and Wm,p(x)(Ω) J. Math. Anal. Appl. 263 424 - 446    DOI : 10.1006/jmaa.2000.7617
Fan X. , Shen J. , Zhao D. (2001) Sobolev embedding theorems for spaces Wk,p(x)(Ω) J. Math. Anal. Appl. 262 749 - 760    DOI : 10.1006/jmaa.2001.7618
Fan X. L. , Zhang Q. H. (2003) Existence of solutions for p(x)-Laplacian Dirichlet problem Nonlinear Anal. 52 1843 - 1852    DOI : 10.1016/S0362-546X(02)00150-5
Fan X. L. , Zhang Q. H. , Zhao D. (2005) Eigenvalues of p(x)-Laplacian Dirichlet problem J. Math. Anal. Appl. 302 306 - 317    DOI : 10.1016/j.jmaa.2003.11.020
Fan X. L. , Han X. Y. (2004) Existence and multiplicity of solutions for p(x)-Laplacian equations in ℝN Nonlinear Anal. 59 173 - 188
Fan X. L. (2007) Global C1, αregularity for variable exponent elliptic equations in divergence form J. Differential Equations 235 397 - 417    DOI : 10.1016/j.jde.2007.01.008
Fan X. L. , Zhao D. (1998) On the generalized Orlicz-Sobolev space Wk,p(x)(Ω) J. Gansu Educ. College 12 (1) 1 - 6
Fernández Bonder J. , Silva A. The concentration-compactness principle for variable exponent spaces and applications, arXiv: 0906. 1992v2 [Math.AP].
Fu Y. (2009) The principle of concentration compactness in Lp(x)(Ω) spaces and its application Nonlinear Anal. 71 1876 - 1892    DOI : 10.1016/j.na.2009.01.023
Zhang X. , Fu Y. Q. (2010) Multiple solutions for a class of p(x)-Laplacian equations involving the critical exponent Ann. Polon. Math. 98 91 - 102    DOI : 10.4064/ap98-1-6
Fu Y. Q. , Zhang X. (2010) Multiple solutions for a class of p(x)-Laplacian equations in ℝninvolving the critical exponent Proc. R. Soc. Lond. Ser. A 466 1667 - 1686    DOI : 10.1098/rspa.2009.0463
Chabrowski J. , Fu Y. Q. (2005) Existence of Solutions for p(x)-Laplacian problems on a bounded domain J. Math. Anal. Appl. 306 604 - 618    DOI : 10.1016/j.jmaa.2004.10.028
Fu Y. Q. (2007) Existence of solutions for p(x)-Laplacian problem on an unbounded domain Topol. Methods Nonlinear Anal. 30 235 - 249
Ghoussoub N. , Yuan C. (2002) Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents Trans. Am. Math. Soc. 352 5703 - 5743    DOI : 10.1090/S0002-9947-00-02560-5
Halsey T. C. (1992) Electrorheological fluids Science 258 761 - 766    DOI : 10.1126/science.258.5083.761
He X. M. , Zou W. M. (2009) Infinitely many arbitrarily small solutions for sigular elliptic problems with critical Sobolev-Hardy exponents Proc. Edinburgh Math. Society 52 97 - 108    DOI : 10.1017/S0013091506001568
Kováčik O. , Rákosnik J. (1991) On spaces Lp(x)and W1,p(x) Czech. Math. J. 41 592 - 618
Kajikiya R. (2005) A critical-point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations J. Funct. Analysis 225 352 - 370    DOI : 10.1016/j.jfa.2005.04.005
Li S. , Zou W. (1998) Remarks on a class of elliptic problems with critical exponents Nonlin. Analysis 32 769 - 774    DOI : 10.1016/S0362-546X(97)00499-9
Li G.B. , Zhang G. (2009) Multiple solutions for the p&q-Laplacian problem with critical exponent Acta Math. Scientia 29B 903 - 918
Lions P. L. (1985) The concentration-compactness principle in the caculus of variation: the limit case, I Rev. Mat. Ibero. 1 45 - 120    DOI : 10.4171/RMI/12
Lions P. L. (1985) The concentration-compactness principle in the caculus of variation: the limit case, II Rev. Mat. Ibero. 1 145 - 201    DOI : 10.4171/RMI/6
Rabinowitz P. H. 1986 Minimax methods in critical-point theory with applications to differential equations CBME Regional Conference Series in Mathematics American Mathematical Society Providence, RI 65
Ruzicka M. 2002 Electrorheological Fluids Modeling and Mathematical Theory Springer-Verlag Berlin
Sharapudinov I. (1978) On the topology of the space Lp(t)([0; 1]) Matem. Zametki 26 613 - 632
Zhikov V. (1987) A veraging of functionals in the calculus of variations and elasticity Math. USSR Izv. 29 33 - 66    DOI : 10.1070/IM1987v029n01ABEH000958
Zhang Q.H. (2006) Existence of radial solutions for p(x)-Laplacian equations in ℝN J. Math. Anal. Appl. 315 (2) 506 - 516    DOI : 10.1016/j.jmaa.2005.10.003