A critical nonlinear fractional elliptic equation with saddle-like potentical in $\mathbb{R}^N$

Abstract

In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ ε^{2s} (-Δ)^{s}{u}+V(z)u=λ|u|^{q-2}u+|u|^{2^{*}_{s}-2}u\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $ε, λ>0$ are positive parameters, $q \in (2,2^{*}_{s}), 2^{*}_{s}=\frac{2N}{N-2s}, $ $N > 2s,$ $s \in (0,1),$ $ (-Δ)^{s}u$ is the fractional laplacian, and $V$ is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Benci and Cerami.