Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents

Pages: 695 - 707, Issue special, November 2013

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Inbo Sim - Department of Mathematics, University of Ulsan, Ulsan 680-749, South Korea (email)
Yun-Ho Kim - Department of Mathematics Education, Sangmyung University, Seoul 110-743, South Korea (email)

Abstract: We study the following nonlinear problem \begin{equation*} -div(w(x)|\nabla u|^{p(x)-2}\nabla u)=\lambda f(x,u)\quad in \Omega \end{equation*} which is subject to Dirichlet boundary condition. Under suitable conditions on $w$ and $f$, employing the variational methods, we show the existence of solutions for the above problem in the weighted variable exponent Lebesgue-Sobolev spaces. Also we obtain the positivity of the infimum eigenvalue for the problem.

Keywords:  $p(x)$-Laplacian, weighted variable exponent Lebesgue-Sobolev spaces, mountain pass theorem, fountain theorem, eigenvalue.
Mathematics Subject Classification:  35J20, 35J60, 35J70, 47J10, 46E35.

Received: August 2012;      Revised: March 2013;      Published: November 2013.