On the superlinear problem involving the $p(x)$-Laplacian - Chao Ji (Electronic Journal of Qualitative Theory of Differential Equations, 2011)
This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: \begin{align*}\left\{ \begin{aligned} &-div(|\nabla u|^{p(x)-2}\nabla u)=\lambda f(x,u) \quad \text{in } \Omega,\\ &u=0 \quad \text{on } \partial \Omega, \end{aligned} \right.\end{align*} where $\Omega\subset R^{N}(N\geq 2)$ is a bounded domain with smooth boundary $\partial \Omega$, $\lambda>0$ is a parameter. Existence of nontrivial solution is established for arbitrary $\lambda>0$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter $\lambda>0$. Then, it is considered the continuation of the solutions. Our results are a generalization of Miyagaki and Souto.