Infinitely many positive solutions for nonlinear fractional Schr\"{o}dinger equations

TL;DR

This paper proves the existence of infinitely many positive solutions for a class of nonlinear fractional Schrödinger equations with radial potential functions, under certain asymptotic conditions at infinity.

Contribution

It establishes the existence of infinitely many non-radial positive solutions for fractional Schrödinger equations with specific potential decay conditions, expanding the understanding of solution multiplicity.

Findings

Existence of infinitely many positive solutions.

Solutions can have arbitrarily large energy.

Solutions are non-radial and positive.

Abstract

We consider the following nonlinear fractional Schr\"{o}dinger equation $$ (-\Delta)^su+u=K(|x|)u^p,\ \ u>0 \ \ \hbox{in}\ \ R^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.