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

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.