Korean J. Math.  Vol 22, No 3 (2014)  pp.395-406
DOI: https://doi.org/10.11568/kjm.2014.22.3.395

Existence of a positive infimum eigenvalue for the $p(x)$-Laplacian Neumann problems with weighted functions

Yun-Ho Kim

Abstract


We study the following nonlinear problem
\begin{equation*}
-\text{div}(w(x)|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u=\lambda f(x,u) \quad \text{in } \Omega
\end{equation*}
which is subject to Neumann boundary condition. Under suitable conditions on $w$ and $f$, we give the existence of a positive infimum eigenvalue for the $p(x)$-Laplacian Neumann problem.

Keywords


$p(x)$-Laplacian; Neumann boundary condition; Weighted variable exponent Lebesgue-Sobolev spaces; Weak solution; Eigenvalue

Subject classification

35D30; 35J60; 35J92; 35P30; 47J10

Sponsor(s)

This research was supported by a 2012 Research Grant from Sangmyung University.

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