On Concentration of Positive Bound States of Nonlinear Schrödinger Equations with Competing Potential Functions

Abstract

Some nonlinear Schrödinger equations with several competing potential functions are considered. Ground states (least energy solutions) are proved to exist and concentrate at a point in the semiclassical limit. The concentration points are shown to be located on the middle ground of the competing potential functions and in some cases are given explicitly in terms of these functions. Also given is a necessary condition for location of concentration of positive bound states (solutions with higher but finite energy).

Keywords

This publication has 18 references indexed in Scilit:

Homoclinic type solutions for a semilinear elliptic PDE on ℝⁿ
Communications on Pure and Applied Mathematics, 1992
Uniqueness of the ground state solutions of △u+f(u)=0 in Rⁿ, n≥3
Communications in Partial Differential Equations, 1991
Uniqueness of positive solutions of Δu−u+up=0 in Rn
Archive for Rational Mechanics and Analysis, 1989
Critical Point Theory and Hamiltonian Systems
Published by Springer Nature ,1989
Remarks on a semilinear elliptic equation on Rn
Journal of Differential Equations, 1988
On the existence of positive entire solutions of a semilinear elliptic equation
Archive for Rational Mechanics and Analysis, 1986
Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential
Journal of Functional Analysis, 1986
The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1.
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 1984
Nonlinear scalar field equations, I existence of a ground state
Archive for Rational Mechanics and Analysis, 1983
Action minima among solutions to a class of Euclidean scalar field equations
Communications in Mathematical Physics, 1978