Existence and concentration of bound states of nonlinear Schrödinger equations with compactly supported and competing potentials

Mingwen Fei and Huicheng Yin

Vol. 244 (2010), No. 2, 261–296

DOI: 10.2140/pjm.2010.244.261

Abstract

We study the existence and concentration of solutions to the N-dimensional nonlinear Schrödinger equation

with u_𝜀(x) > 0 and u_𝜀 ∈ H¹(ℝ^N), where N ≥ 3,  1 < q < p < (N+2)∕(N−2), and 𝜀 > 0 is sufficiently small. We take potential functions V (x) ∈ C₀^∞(ℝ^N) with V (x)≢0 and V (x) ≥ 0, and show that if K(x) and Q(x) are permitted to be unbounded under some necessary restrictions, then a positive solution u_𝜀(x) exists in H¹(ℝ^N) when the corresponding ground energy function G(x) has local minimum points. We establish the concentration property of u_𝜀(x) as 𝜀 tends to zero. We have removed from some previous papers the crucial restriction that the nonnegative potential function V (x) has a positive lower bound or decays at infinity like (1 + |x|)^−α with 0 < α ≤ 2.

Keywords

nonlinear Schrödinger equation, bound state, ground energy function, competing potential, Harnack inequality, concentration and compactness

Mathematical Subject Classification 2000

Primary: 35J10

Secondary: 35J60

Milestones

Received: 1 March 2009

Revised: 2 September 2009

Accepted: 22 September 2009

Published: 14 December 2009