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

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.