Bulletin of the
Korean Mathematical Society BKMS

This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: $$\left\{ \begin{array}{ll} -{\rm{div}}\left((1+\frac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}})|\nabla u|^{p(x)-2}\nabla u\right)= \lambda f(x,u),\;{\rm{a.e.}}\;\;{\rm in}\; ~\Omega, &\\ u=0,\;{\rm{on}} ~\partial \Omega,& \\ \end{array} \right.$$ where $\Omega\subset{\rm{ R}}^{N}$ is a bounded domain with smooth boundary $\partial \Omega$, $\lambda>0$ is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter $\lambda$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter $\lambda>0$. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.

Keywords: superlinear problem, $p(x)$-Laplacian, variational method, variable exponent Sobolev space

MSC numbers: Primary 35D05; Secondary 35J70