Existence of positive solutions for a semipositone p(⋅)-Laplacian problem

Abstract

In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem$\{\begin{array}{cc}-{\Delta }_{p(\cdot )}u=f\left(u\right)-\lambda & \text{in }\Omega \\ u>0& \text{in }\Omega \\ u=0& \text{on }\partial \Omega \end{array},$ where $\Omega \subset R^N$, $N\ge 2$ is a smooth bounded domain, f is a continuous function with subcritical growth, $\lambda >0$ and ${\Delta }_{p(\cdot )}u=\text{div}\left({\left|\nabla u\right|}^{p(\cdot )-2}\nabla u\right)$. Also, we assume an Ambrosetti-Rabinowitz type of condition and using the Mountain Pass arguments, comparison principles and regularity principles we prove the existence of positive weak solution for λ small enough.