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

Abstract

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem \[ \displaystyle \left\{\begin{array}{rcll} (-Δ)_p^s(u) &=& λf(u) \qquad & \text{in} \ \ Ω \\u &=& 0 & \text{in} \ \ \mathbb{R}^N -Ω, \end{array}\right. \] whenever $\lambda >0$ is a sufficiently small parameter. Here $\Omega \subseteq {\mathbb{R}}^N$ a bounded domain with $C^{1,1}$ boundary, $2⩽p<N$, $s\in (0,1)$ 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.