The Brezis–Nirenberg problem for the fractional Laplacian with mixed Dirichlet–Neumann boundary conditions

Abstract

In this work we study the existence of solutions to the critical Brezis–Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet–Neumann boundary conditions, i.e.,{(−Δ)su=λu+u2s⁎−1,u>0inΩ,u=0onΣD,∂u∂ν=0onΣN, where Ω⊂RN is a regular bounded domain, 12<s<1, 2s⁎ is the critical fractional Sobolev exponent, 0≤λ∈R, 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∩Σ‾N=Γ is a smooth (N−2)-dimensional submanifold of ∂Ω.