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

Dongdong Qin; Xianhua Tang; Qingfang Wu

doi:10.3934/cpaa.2019061

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2019, Volume 18, Issue 3: 1261-1280. Doi: 10.3934/cpaa.2019061

This issue Previous Article Symmetry of solutions to a class of Monge-Ampère equations Next Article Stability and

$ L^{p}$

convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

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

1.
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China
2.
School of Traffic and Transportation Engineering, Central South University, Changsha, 410075 Hunan, China

* Corresponding author
* Corresponding author

Received: April 30, 2018

Revised: August 31, 2018

Published: May 2019

This work is partially supported by the National Natural Science Foundation of China (Nos.: 11801574, 11571370, 11501190) of China

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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.

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Mathematics Subject Classification: 35J50; 35J47.

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