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

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., $$ \left\{\begin{array}{rcl} (-\Delta)^su & = & \lambda u+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad \Omega,\\ u & = & 0\quad\mbox{on}\quad \Sigma_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial \nu} & = & 0\quad\mbox{on}\quad \Sigma_{\mathcal{N}}, \end{array}\right. $$ where $\Omega\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ 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 $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$, and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$-dimensional submanifold of $\partial\Omega$.