On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

Authors:	Mihai Mihailescu Vicentiu Radulescu

Institution:	Department of Mathematics, University of Craiova, 200585 Craiova, Romania ; Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Abstract:	We consider the nonlinear eigenvalue problem $\displaystyle -{\rm div}\left(\vert\nabla u\vert^{p(x)-2}\nabla u\right)=\lambda \vert u\vert^{q(x)-2}u$ in $\Omega$ , on $\partial\Omega$ , where $\Omega$ is a bounded open set in $\mathbb{R}^N$ with smooth boundary and , are continuous functions on $\overline\Omega$ such that $1<\inf_\Omega q< \inf_\Omega p<\sup_\Omega q$ , $\sup_\Omega p<N$ , and $q(x)<Np(x)/\left(N-p(x)\right)$ for all $x\in\overline\Omega$ . The main result of this paper establishes that any $\lambda>0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

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