Infinitely many positive standing waves for Schr\"odinger equations with competing coefficients

TL;DR

This paper proves the existence of infinitely many positive standing wave solutions for a class of Schrödinger equations with competing coefficients, extending known results by considering the effects of a nonzero term with specific decay properties.

Contribution

It introduces conditions under which the presence of a nonzero term with competing coefficients guarantees infinitely many positive solutions, a significant extension over previous finite-solution results.

Findings

Infinitely many positive solutions exist under certain decay conditions.

The nonzero term with competing coefficients is crucial for multiplicity.

Results extend the understanding of solution multiplicity in Schrödinger equations.

Abstract

The paper deals with the equation $-\Delta u+a(x) u +b(x)u^q -u^p = 0$, $u \in H^1(\R^N)$, whith $N\ge 2$, $1<q<p,\ p<{N+2\over N-2}$ if $N\ge 3$, $\inf a>0$, $a(x)\to a_\infty$ and $b(x)\to 0$ as $|x|\to\infty$. When $a(x)\le a_\infty$ and $b(x) = 0$ only a finite number of positive solutions to the problem is reasonably expected. Here we prove that the presence of a nonzero term $b(x)u^q $ with $b(x)\geq 0, \ b(x)\neq 0,$ under suitable assumptions on the decay rates of $a$ and $b,$ allows to obtain infinitely many positive solutions.