Existence results for fractional Brezis-Nirenberg type problems in unbounded domains - Shen, Yansheng; Wang, Xumin (Topological Methods in Nonlinear Analysis, 2022)
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 fractionalPoincaré inequality together with the concentration-compactness principlefor fractional Sobolev spaces in unbounded domains, we prove an existenceresult to the equation.