Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

Article type: Research Article

Authors: Chen, Taiyong^a | Jiang, Yahui^a | Squassina, Marco^{b; c; *} | Zhang, Jianjun^d

Affiliations: [a] School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China | [b] College of Science, Princess Nourah Bint Abdul Rahman University, Riyadh, PO Box 84428, Saudi Arabia | [c] Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, Brescia, Italy | [d] College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper, we are concerned with the coupled nonlinear Schrödinger system −ε2Δu+a(x)u=μ1u3+βv2uin RN,−ε2Δv+b(x)v=μ2v3+βu2vin RN, where 1⩽N⩽3, μ1,μ2,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.

Keywords: Nonlinear Schrödinger systems, semiclassical limit, critical frequency, variational methods

DOI: 10.3233/ASY-231828

Journal: Asymptotic Analysis, vol. 134, no. 1-2, pp. 127-154, 2023