Normalized multi-bump solutions for saturable Schrödinger equations

Abstract

In this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function:-Δv+ΓI(ϵx)+v21+I(ϵx)+v2v=λv2/1+1(epsi;x)+v2v =λ v,x ∈ R2. We prove that, with the density function being radially symmetric, for given integer k ≥ 2 there exist a family of non-radial, k-bump type normalized solutions (i.e., with the L2 constraint) which concentrate at the global maximum points of density functions when ϵ → 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.