Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ground state of indefinite coupled nonlinear Schrödinger systems

Ruijin Xu, Jiabao Su, Rushun Tian

Abstract

In this paper, we study the ground state solutions of the following coupled nonlinear Schrödinger system (P) $-Δu_1-τ_1 u_1 =μ_1u_1^3+βu_1u_2^2$, $ -Δu_2-τ_2 u_2 =μ_2u_2^3+βu_1^2u_2$ in $Ω$, $u_1=u_2=0$ on $\partialΩ$, where $μ_1, μ_2>0$, $β>0$ and $Ω\subset \mathbb{R}^N (N\le3)$ is a bounded domain with smooth boundary. We are concerned with the indefinite case, i.e., $τ_1, τ_2$ are greater than or equal to the principal eigenvalue of $-Δ$ with the Dirichlet boundary datum. By delicate variational arguments, we obtain the existence of ground state solution to $(P)$, and also provide information on critical energy levels for coupling parameter $β$ in some ranges.

Ground state of indefinite coupled nonlinear Schrödinger systems

Abstract

In this paper, we study the ground state solutions of the following coupled nonlinear Schrödinger system (P) , in , on , where , and is a bounded domain with smooth boundary. We are concerned with the indefinite case, i.e., are greater than or equal to the principal eigenvalue of with the Dirichlet boundary datum. By delicate variational arguments, we obtain the existence of ground state solution to , and also provide information on critical energy levels for coupling parameter in some ranges.
Paper Structure (5 sections, 17 theorems, 158 equations)

This paper contains 5 sections, 17 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Existence of fully nontrivial ground state solution
  4. Further study on the case $\tau_1=\tau_2=\lambda_1$
  5. Acknowledgments

Key Result

Theorem 1.1

Assume that $\beta> \Lambda:=\max \left\{\hat{\beta}_1, \hat{\beta}_2\right\}$. Then the system sy1 has a ground state solution $\mathbf{u^*}$. Moreover, it holds that in which $e$, $c'$ and $c_{sem}$ are defined in eq:groundEnergy, eq:level_cprime and eq:level_csem, respectively, and where and $\rho(\mathbf{u})>0$ will be defined in Lemma l-g.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • ...and 26 more