Ground state solutions for a critical fractional Laplacian equation in unbounded domains: existence and regularity - Shen, Yansheng (Topological Methods in Nonlinear Analysis, 2024)
In this work we study the following fractional critical problem align* cases (-Δ)^su=λ|u|^q-2u+|u|^2_s^*-2u & in Ω, u=0 & in R^NΩ, cases align* where Ω is an open unbounded strip-like domain in R^N, N> 2s with s(0,1), λ> 0, q[2,2_s^*) and 2_s^*=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^∞-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.