Existence of positive solutions for a semipositone $p(\cdot )$-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 \[ \left\{ \begin{array}{cc} -Δ_{p(\cdot )}u=f(u)-λ& \text{in }Ω\\ u>0 & \text{in }Ω\\ u=0 & \text{on }\partial Ω\end{array}% \right. , \] where $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$ is a smooth bounded domain, $f$ is a contiuous function with subcritical growth, $\lambda >0$ and ${\Delta }_{p(\cdot )}u=\text{div}({\left|\nabla u\right|}^{p(\cdot )-2}\nabla u)$. Also, we assume an Ambrosetti-Rabinowitz type of condition and using the Mountain Pass arguments, comparision principles and regularity principles we prove the existence of positive weak solution for $\lambda$ small enough.