Existence results for fractional Brezis-Nirenberg type problems in unbounded domains

Abstract

In this paper we study the fractional Brezis-Nirenberg type problems in unbounded cylinder-type domains \begin{align*} \begin{cases} (-\Delta)^{s}u-\mu\dfrac{u}{|x|^{2s}}=\lambda u+|u|^{2^{\ast}_{s}-2}u & \text{in } \Omega,\\ u=0 & \text{in } \mathbb{R}^{N}\setminus \Omega, \end{cases} \end{align*} where $(-\Delta)^{s}$ is the fractional Laplace operator with $s\in(0,1)$, $\mu\in[0,\Lambda_{N,s})$ with $\Lambda_{N,s}$ the best fractional Hardy constant, $\lambda> 0$, $N> 2s$ and $2^{\ast}_{s}={2N}/({N-2s})$ denotes the fractional critical Sobolev exponent. By applying the fractional Poincaré inequality together with the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove an existence result to the equation.