Infinitely many positive standing waves for Schrödinger equations with competing coefficients

Abstract

The paper deals with the equation $-\Delta u+a(x)u+b(x)u^q-u^p=0$, $u\in H^1({\mathbb{R}}^N)$, whith $N\ge 2$, $1<q<p,p<\frac{N+2}{N-2}$ 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 $b(x)\ge 0,b(x)\ne 0,$ under suitable assumptions on the decay rates of $a$ and $b,$ allows to obtain infinitely many positive solutions.