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 \[ \displaystyle \left\{\begin{array}{rcll} (-\Delta)_p^s(u)&=&\lambda f(u) \qquad&\text{in} \ \ \Omega \\u&=&0&\text{in} \ \ \mathbb{R}^N -\Omega , \end{array}\right. \] whenever $\lambda>0$ is a sufficiently small parameter. Here $\Omega \subseteq \mathbb{R}^N$ a bounded domain with $C^{1,1}$ boundary, $2\leqslant p0$ 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.