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 $\lim_{|z|\to∞}\frac{F(x,z)}{|z|^2} = ∞$ uniformly in $x$, we introduce a new local super-quadratic condition that allows the nonlinearity $F$ to be super-quadratic at some $x∈ \mathbb{R}^N$ and asymptotically quadratic at other $x∈ \mathbb{R}^N$. Employing some analytical skills and using the variational method, we prove some results about the existence of ground states for the system with periodic or non-periodic potentials. In particular, any nontrivial solutions are continuous and decay to zero exponentially as $|x| \to ∞$. Our main results extend and improve some recent ones in the literature.