Infinitely many positive solutions for the nonlinear Schrödinger equations

Abstract. We consider the following nonlinear Schrödinger equa-tion { ∆u − (1 + δV)u+ f(u) = 0 in RN, u> 0 in RN, u ∈ H1(RN) where V is a continuous potential and f(u) is a nonlinearity sat-isfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a δ0 such that for 0 < δ < δ0, the above problem has in-finitely many positive solutions. This generalizes and gives a new proof of the results by Cerami-Passaseo-Solimini [13]. The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems. 1