Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

Abstract

We consider the following nonlinear Schrodinger equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta u-(1+\delta V)u+f(u)=0 \ \ \hbox { in }\mathbb {R}^N,\\ u>0 \ \hbox {in} \ \mathbb {R}^N, u\in H^1(\mathbb {R}^N) \end{array} \right. \end{aligned}$$ where $$V$$ is a continuous potential and $$ f(u)$$ is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a $$\delta _0$$ such that for $$0<\delta <\delta _0$$ , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami et al. (Comm. Pure Appl. Math. 66, 372–413, 2013). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.