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., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $\Omega \subset {\mathbb{R}}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^{\ast }$ is the critical fractional Sobolev exponent, $0\le \lambda \in \mathbb{R}$, $\nu$ is the outwards normal to $\partial \Omega$, ${\Sigma }_{\mathcal{D}}$, ${\Sigma }_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partial \Omega$ such that ${\Sigma }_{\mathcal{D}}\cup {\Sigma }_{\mathcal{N}}=\partial \Omega$, ${\Sigma }_{\mathcal{D}}\cap {\Sigma }_{\mathcal{N}}=\varnothing$, and ${\Sigma }_{\mathcal{D}}\cap {\overline{\Sigma }}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$-dimensional submanifold of $\partial \Omega$.