Positive Solutions for a Mixed Local-Nonlocal Problem with Semipositone Nonlinearity

TL;DR

This paper establishes the existence of positive solutions for a mixed local-nonlocal semipositone problem using variational methods, comparison principles, and regularity results.

Contribution

It proves the existence of positive solutions for a new class of mixed local-nonlocal problems with semipositone nonlinearities.

Findings

Existence of at least one positive solution is proven.

The problem involves a combination of local and nonlocal operators.

Variational and comparison techniques are successfully applied.

Abstract

In this article, we prove the existence of at least one positive solution for the mixed local-nonlocal semipositone problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u+ (-\Delta)^s_p u &= \lambda f(u) && \text{in } \Omega, u &= 0 && \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} using mountain pass arguments, comparison principles and regularity principles.