Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem

TL;DR

This paper proves the existence of positive solutions for a semipositone variable exponent p-Laplacian problem using variational methods, under certain growth and regularity conditions, for small positive parameters.

Contribution

It establishes the existence of positive weak solutions for a semipositone p(x)-Laplacian problem with variable exponent, extending previous results to this more general setting.

Findings

Positive weak solutions exist for small λ.

Solutions are obtained using Mountain Pass theorem.

Conditions on f ensure subcritical growth and applicability of variational methods.

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 \[ \left\{ \begin{array}{cc} -\Delta _{p(\cdot )}u=f(u)-\lambda & \text{in }\Omega \\ u>0 & \text{in }\Omega \\ u=0 & \text{on }\partial \Omega \end{array}% \right. , \] where $\Omega \subset \mathbb{R}^{N}$, $N\geq 2$ is a smooth bounded domain, $f$ is a contiuous function with subcritical growth, $\lambda >0$ and $\Delta _{p(\cdot )}u=\text{div}(\left\vert \nabla u\right\vert ^{p(\cdot )-2}\nabla u)$. 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.