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

We consider the following nonlinear Schrodinger equation [{l} Δu-(1+δV)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $f(u)$ is a superlinear nonlinearity satisfying some nondegeneracy condition. Using localized energy method, we prove that there exists some $δ_0$ such that for $0<δ<δ_0$, the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami-Passaseo-Solimini (CPAM to appear). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.

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