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 $\partial\Omega$.
Categories: math.AP