Ground state solutions for fractional scalar field equations under a general critical nonlinearity

TL;DR

This paper investigates the existence of ground state solutions for fractional scalar field equations involving the fractional Laplacian with general nonlinearities that include critical growth, covering both polynomial and exponential cases.

Contribution

It establishes existence results for ground states in fractional equations with broad nonlinearities, including critical growth, for all dimensions and fractional orders.

Findings

Existence of ground state solutions for fractional equations with critical nonlinearities.

Results cover both polynomial and exponential growth cases.

Applicable to all dimensions with fractional order in (0,1).

Abstract

In this paper we study existence of ground state solution to the following problem $$ (- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N) $$ where $(-\Delta)^{\alpha}$ is the fractional Laplacian, $\alpha\in (0,1)$. We treat both cases $N\geq2$ and $N=1$ with $\alpha=1/2$. The function $g$ is a general nonlinearity of Berestycki-Lions type which is allowed to have critical growth: polynomial in case $N\geq2$, exponential if $N=1$.