Infinitely many positive solutions for the nonlinear Shcrodinger equations in $R^N$

TL;DR

This paper proves the existence of infinitely many non-radial positive solutions to a nonlinear Schrödinger equation in R^N, under specific asymptotic conditions on the potential function V(r).

Contribution

It establishes the existence of infinitely many solutions with arbitrarily large energy for a class of nonlinear Schrödinger equations with particular potential decay.

Findings

Existence of infinitely many non-radial solutions.

Solutions' energy can be arbitrarily large.

Conditions on V(r) ensure solution multiplicity.

Abstract

We consider the following nonlinear problem in $\R^N$ $$\label{eq} - \Delta u +V(|y|)u=u^{p},\quad u>0 {in} \R^N, u \in H^1(\R^N) $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that if $V(r)$ has the following expansion: There are constants $a>0$, $m>1$, $\theta>0$, and $V_0>0$, such that \[ V(r)= V_0+\frac a {r^m} +O\bigl(\frac1{r^{m+\theta}}\bigr),\quad \text{as $r\to +\infty$,} \] then \eqref{eq} has {\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large.