Abstract

Let V (x) be a non-negative, bounded potential in RN, N ≥ 3 and p supercritical, p > frac(N + 2, N - 2). We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δ u - V (x) u + up = 0  in  RN, with u (x) → 0 as | x | → + ∞. We prove that if V (x) = o (| x |-2) as | x | → + ∞, then for N ≥ 4 and p > frac(N + 1, N - 3) this problem admits a continuum of solutions. If in addition we have, for instance, V (x) = O (| x |- μ) with μ > N, then this result still holds provided that N ≥ 3 and p > frac(N + 2, N - 2). Other conditions for solvability, involving behavior of V at ∞, are also provided. © 2007 Elsevier Inc. All rights reserved.