Variational analysis for an indefinite quasilinear problem with variable exponent

被引:1
|
作者
Bouslimi, Mabrouk [1 ]
Kefi, Khaled [2 ]
Preda, Felician-Dumitru [3 ]
机构
[1] Univ Tunis, Inst Preparatoire Etud Ingn Tunis, Rue Jawaher Lel Nehru, Tunis 1008, Tunisia
[2] Univ Sousse, Inst Super Transport & Logist Sousse, Sousse 4029, Tunisia
[3] Inst Math Stat & Appl Math, Bucharest 050711, Romania
关键词
p(x)-Laplace operator; generalized Sobolev spaces; mountain pass theorem; Ekeland's variational principle; weak solution;
D O I
10.1515/APAM.2011.011
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study the nonlinear boundary value problem -div((vertical bar del u(x)vertical bar(p1(x)-2) + vertical bar del u(x)vertical bar(p2(x)-2))del u(x)) =lambda V-1(x)vertical bar u vertical bar(q(x)-2)u - mu V-2(x)vertical bar u vertical bar(alpha(x)-2)u in Omega, u = 0 on delta Omega, where Omega is a bounded domain in R-N with smooth boundary, lambda, mu are positive real numbers, p(1), p(2), q, alpha are continuous functions on Omega, V-1 and V-2 are weight functions in generalized Lebesgue spaces L-s1(x)(Omega) and L-s2(x)(Omega), respectively, such that V-1 > 0 in an open set Omega(0) subset of Omega with |Omega(0|) > 0 and V-2 >= 0 on Omega. We prove, under appropriate conditions that for any mu > 0 there exists lambda(*) sufficiently small such that for any lambda is an element of(0,lambda(*)) the above nonhomogeneous quasilinear problem has a nontrivial positive weak solution. The proof relies on some variational arguments based on Ekeland's variational principle.
引用
收藏
页码:67 / 83
页数:17
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