# Existence and symmetry of solutions for critical fractional Schr\"odinger equations with bounded potentials

## TL;DR

This paper proves the existence of radially symmetric solutions for critical fractional Schrödinger equations with bounded potentials using concentration compactness and minimax methods, without requiring the Ambrosetti-Rabinowitz condition.

## Contribution

It establishes the existence of solutions for critical fractional Schrödinger equations under new conditions, removing the need for the Ambrosetti-Rabinowitz growth condition.

## Key findings

- Existence of nontrivial radially symmetric solutions
- Solutions obtained without Ambrosetti-Rabinowitz condition
- Application of concentration compactness in fractional Sobolev spaces

## Abstract

This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N}, \end{eqnarray*} where $(-\Delta)^{\alpha}$ is the fractional Laplacian operator with $\alpha\in(0,1)$, $N\geq2$, $\lambda$ is a positive real parameter and $2_{\alpha}^{*}=2N/(N-2\alpha)$ is the critical Sobolev exponent, $V(x)$ and $k(x)$ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the subcritical nonlinearity.