Existence of positive solutions for a parameter fractional $p$-Laplacian problem with semipositone nonlinearity (2211.02790v1)

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.