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 {(−Δ)ps(u)=λf(u)inΩu=0inRN−Ω, whenever λ>0 is a sufficiently small parameter. Here Ω⊆RN a bounded domain with C1,1 boundary, 2≤p0 is chosen sufficiently small the associated Energy Functional to the problem has a mountain pass structure and, therefore, it has a critical point uλ, 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.