Existence and decay property of ground state solutions for Hamiltonian elliptic system

Jian Zhang; Wen Zhang

doi:10.3934/cpaa.2019110

Article Contents

2019, Volume 18, Issue 5: 2433-2455. Doi: 10.3934/cpaa.2019110

This issue Previous Article Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise Next Article Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian

Existence and decay property of ground state solutions for Hamiltonian elliptic system

Jian Zhang^1,2, and
Wen Zhang^1,2, ,

1.
School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China
2.
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

^* Corresponding author
^* Corresponding author

Received: April 30, 2018

Revised: November 30, 2018

Published: March 2019

This work was supported by the NNSF (Nos. 11701173, 11601145, 11571370, 61772196), by the Natural Science Foundation of Hunan Province (Nos. 2017JJ3130, 2017JJ3131), by the Excellent youth project of Education Department of Hunan Province (17B143), by the Hunan University of Commerce Innovation Driven Project for Young Teacher (16QD008), and by the Project funded by China Postdoctoral Science Foundation (2018M640758)

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Abstract

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term

$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u +\vec{b}(x)\cdot \nabla u +u+V(x)v = f(x, |z|)v, \; \; x\in\mathbb{R}^{N}, \\ -\Delta v -\vec{b}(x)\cdot \nabla v +v+V(x)u = f(x, |z|)u, \; \; x\in\mathbb{R}^{N}, \ \end{array} \right. \end{eqnarray*} $

where $ z = (u, v)\in\mathbb{R}^{2} $. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground state solutions, such as compactness of set of ground state solutions and exponential decay of ground state solutions.

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Mathematics Subject Classification: 35J50, 58E05.

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