Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

Author:

Chen Taiyong1,Jiang Yahui1,Squassina Marco23,Zhang Jianjun4

Affiliation:

1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China

2. College of Science, Princess Nourah Bint Abdul Rahman University, Riyadh, PO Box 84428, Saudi Arabia

3. Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, Brescia, Italy

4. College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

Abstract

In this paper, we are concerned with the coupled nonlinear Schrödinger system − ε 2 Δ u + a ( x ) u = μ 1 u 3 + β v 2 u in  R N , − ε 2 Δ v + b ( x ) v = μ 2 v 3 + β u 2 v in  R N , where 1 ⩽ N ⩽ 3, μ 1 , μ 2 , β > 0, a ( x ) and b ( x ) are nonnegative continuous potentials, and ε > 0 is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as ε → 0, when a ( x ) and b ( x ) achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.

Publisher

IOS Press

Subject

General Mathematics

Reference33 articles.

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4. A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system;Bartsch;Calc. Var. PDE.,2010

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