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

TL;DR

This paper investigates the existence of solutions to a critical fractional Laplacian problem with mixed boundary conditions, extending classical results to fractional operators and complex boundary geometries.

Contribution

It introduces new existence results for the Brezis-Nirenberg problem involving the spectral fractional Laplacian with mixed Dirichlet-Neumann boundary conditions.

Findings

Existence of solutions for certain parameter ranges.

Extension of classical results to fractional and mixed boundary conditions.

Analysis of boundary geometry effects on solutions.

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., $$ \left\{\begin{array}{rcl} (-\Delta)^su & = & \lambda u+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad \Omega,\\ u & = & 0\quad\mbox{on}\quad \Sigma_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial \nu} & = & 0\quad\mbox{on}\quad \Sigma_{\mathcal{N}}, \end{array}\right. $$ 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 $\partial\Omega$.