Existence of positive solutions for a parameter fractional $p$-Laplacian problem with semipositone nonlinearity

TL;DR

This paper establishes the existence of positive solutions for a fractional p-Laplacian problem with semipositone nonlinearity, using variational methods and boundary regularity results for small parameters.

Contribution

It demonstrates the existence of positive solutions for a nonlocal fractional p-Laplacian problem with semipositone nonlinearity, employing mountain pass techniques and boundary regularity.

Findings

Existence of at least one positive solution for small .

The associated energy functional has a mountain pass structure.

Solutions are shown to be positive using regularity and Hopf's Lemma.

Abstract

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-\Delta)_p^s(u) &=& \lambda f(u) \qquad & \text{in} \ \ \Omega \\u &=& 0 & \text{in} \ \ \mathbb{R}^N -\Omega , \end{array}\right. \] whenever $\lambda >0$ is a sufficiently small parameter. Here $\Omega \subseteq \mathbb{R}^N$ a bounded domain with $C^{1,1}$ boundary, $2\leqslant p <N$, $s\in (0,1)$ and $f$ superlineal and subcritical. We prove that if $\lambda>0$ is chosen sufficiently small the associated Energy Functional to the problem has a mountain pass structure and, therefore, it has a critical point $u_\lambda$, which is a weak solution. After that we manage to prove that this solution is positive by using new regularity results up to the boundary and a Hopf's Lemma.