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

In this paper we prove the existence of at least one positive solution for the nonlocal semipositone problem $${\displaystyle \{\begin{array}{rcll}(-\mathrm{\Delta }{)}_p^s(u)& =& \lambda f(u)& \text{in }\mathrm{\Omega }\\ u& =& 0& \text{in }{\mathbb{R}}^N-\mathrm{\Omega },\end{array}}$$ whenever $\lambda >0$ is a sufficiently small parameter. Here $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ 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.