Ground state solutions for a critical fractional Laplacian equation in unbounded domains: existence and regularity

Abstract

In this work we study the following fractional critical problem \begin{align*} \begin{cases} (-\Delta)^{s}u=\lambda|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u & \mbox{in } \Omega, \\ u=0 & \text{in }\mathbb{R}^{N}\setminus \Omega, \end{cases} \end{align*} where $\Omega$ is an open unbounded strip-like domain in $\mathbb{R}^{N}$, $N> 2s$ with $s\in(0,1)$, $\lambda> 0$, $q\in[2,2_{s}^{\ast})$ and $2_{s}^{\ast}=2N/(N-2s)$. By variational methods, we prove the existence of positive ground state solutions to the problem. Further, we study the regularity of these solutions. Precisely, using a Brézis-Kato type estimate for unbounded domains, we establish the $L^{\infty}$-bound on nonnegative solutions of the equation for certain range of $q$. The present work extends the existence and regularity results for fractional Laplace equations to unbounded domains.