Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials

Abstract

In this paper we study a class of weakly coupled Schrödinger system arising in several branches of sciences, such as nonlinear optics and Bose-Einstein condensates. Instead of the well known super-quadratic condition that lim |z|→∞ F (x,z) |z| 2 1. Introduction. We consider following coupled gradient system of nonlinear Schrödinger equations:

Ground state solutions for Hamiltonian elliptic systems with super or asymptotically quadratic nonlinearity

“…Afterwards, condition (SQ) was used in many papers, see [13,14,26,33,38,42,47] and the references therein. In a recent paper [30], a local superquadratic condition instead of (SQ) was introduced by Qin, Tang, and Wu, which combines with a technical condition to guarantee the existence of ground state solutions for Schrödinger systems like (1.1). Note that the assumptionĜ(x, η) > 0 in (DL), (G 3 ), and (G 4 ) plays an important role in the arguments of Zhang [40] and Zhao [46], see also Ding and Lee [13].…”