Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2016; 53(6): 1805-1821

Online first article November 4, 2016      Printed November 30, 2016

https://doi.org/10.4134/BKMS.b151001

Copyright © The Korean Mathematical Society.

Multiple solutions for equations of $p(x)$-Laplace type with nonlinear Neumann boundary condition

Yun-Ho Kim and Kisoeb Park

Sangmyung University, Incheon National University

Abstract

In this paper, we are concerned with the nonlinear elliptic equations of the $p(x)$-Laplace type $$ \begin{cases} -\text{div}(a(x,\nabla u))+|u|^{p(x)-2}u=\lambda f(x,u) \quad &\textmd{in} \quad \Omega \\ a(x,\nabla u)\frac {\partial u}{\partial n} = \lambda\theta g(x,u) &\textmd{on} \quad \partial\Omega, \end{cases} $$ which is subject to nonlinear Neumann boundary condition. Here the function $a(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \overline{\Omega} \to (1,\infty)$ and the functions $f, g$ satisfy a Carath\'eodory condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.

Keywords: three critical points, $p(x)$-Laplacian, variable exponent Lebesgue-Sobolev spaces

MSC numbers: 35D30, 35D50, 35J15, 35J60, 35J62