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

TL;DR

This paper proves the existence of positive solutions for a fractional elliptic equation with saddle-like potential in ^N, using variational methods and minimax techniques.

Contribution

It introduces a new approach to establish positive solutions for fractional elliptic equations with saddle-like potentials via constrained minimization.

Findings

Existence of positive solutions established.

Application of Nehari manifold and minimax methods.

Results valid for a range of parameters psilon, mbda, and q.

Abstract

In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=\lambda |u|^{q-2}u+|u|^{2^{*}_{s}-2}u\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon, \lambda >0$ are positive parameters, $q \in (2,2^{*}_{s}), 2^{*}_{s}=\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.