Concave-Convex critical problems for the spectral fractional Laplacian with mixed boundary conditions

Abstract

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities,N −2s the critical fractional Sobolev exponent, λ > 0, ν is the outwards normal to ∂Ω, ΣD, ΣN are smooth (N − 1)-dimensional submanifolds of ∂Ω such that ΣD ∪ ΣN = ∂Ω, ΣD ∩ ΣN = ∅, and ΣD ∩ ΣN = Γ is a smooth (N − 2)-dimensional submanifold of ∂Ω.In particular, we will prove that, for the sublinear case 0 < q < 1, there exists at least two solutions for every 0 < λ < Λ for certain Λ ∈ R while, for the superlinear case 1 < q < 2 * s − 1, we will prove that there exists at least one solution for every λ > 0. We will also prove that solutions are bounded.