Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

Abstract

We consider the standing-wave problem for a nonlinear Schrödinger equation, corresponding to the semilinear elliptic problem − u + V(x)u = |u|p−1u, u ∈ H1(R2), where V(x) is a uniformly positive potential and p > 1. Assuming that V(x) = V∞ + a |x|m + O

1 |x|m+σ

, as |x| → +∞, for instance if p > 2, m > 2 andσ > 1 we prove the existence of infinitely many positive solutions. If V(x) is radially symmetric, this result was proved in [43]. The proof without symmetries is much more difficult, and for that we develop a new intermediate Lyapunov– Schmidt reductionmethod,which is a compromise between the finite and infinite dimensional versions of it.