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 $\epsilon ,\lambda >0$ are positive parameters, $q\in (2,2_s^{\ast }),2_s^{\ast }=\frac{2N}{N-2s},$ $N>2s,$ $s\in (0,1),$ $(-\Delta {)}^{s}u$ 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.

arXiv admin note: text overlap with arXiv:1506.04947