Bound states of asymptotically linear Schrödinger equations with compactly supported potentials

Mingwen Fei and Huicheng Yin

Vol. 261 (2013), No. 2, 335–367

DOI: 10.2140/pjm.2013.261.335

Abstract

We study the existence and concentration of bound states to N-dimensional nonlinear Schrödinger equation −𝜀²△u_𝜀 + V (x)u_𝜀 = K(x)f(u_𝜀), where N ≥ 3, 𝜀 > 0 is sufficiently small, and the function f(s) is nonnegative and asymptotically linear at infinity. More concretely, when f(s) ∼ O(s) as s → +∞, the potential function V (x) lies in C₀¹(ℝ^N) with V (x) ≥ 0 and V (x)≢0, and K(x) ≥ 0 is permitted to be unbounded under some other necessary restrictions, we can show that a positive H¹(ℝ^N)-solution u_𝜀(x) exists and concentrates around the local maximum point of the corresponding ground energy function.

Keywords

nonlinear Schrödinger equation, bound state, asymptotically linear, Harnack inequality, concentration-compactness

Mathematical Subject Classification 2010

Primary: 35J10, 35J60, 35Q55, 35J91

Milestones

Received: 25 August 2011

Revised: 17 August 2012

Accepted: 16 October 2012

Published: 20 March 2013