On vector solutions for coupled nonlinear Schrödinger equations with critical exponents

Seunghyeok Kim

doi:10.3934/cpaa.2013.12.1259

Article Contents

2013, Volume 12, Issue 3: 1259-1277. Doi: 10.3934/cpaa.2013.12.1259

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On vector solutions for coupled nonlinear Schrödinger equations with critical exponents

Seunghyeok Kim^1,

1.
Department of Mathematics, Pohang University of Science and Technology, Pohang, Kyungbuk

Received: December 2011

Revised: June 2012

Published: May 2013

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Abstract

In this paper, we study the existence and asymptotic behavior of a solution with positive components (which we call a vector solution) for the coupled system of nonlinear Schrödinger equations with doubly critical exponents \begin{eqnarray*} \Delta u + \lambda_1 u + \mu_1 u^{\frac{N+2}{N-2}} + \beta u^{\frac{2}{N-2}}v^{\frac{N}{N-2}} = 0\\ \Delta v + \lambda_2 v + \mu_2 v^{\frac{N+2}{N-2}} + \beta u^{\frac{N}{N-2}}v^{\frac{2}{N-2}} = 0 \quad in \quad \Omega\\ u, v > 0 \quad in \quad \Omega, \quad u, v = 0 \quad on \quad \partial \Omega \end{eqnarray*} as the coupling coefficient $\beta \in R$ tends to 0 or $+\infty$, where the domain $\Omega \subset R^n (N \geq 3)$ is smooth bounded and certain conditions on $\lambda_1, \lambda_2 > 0$ and $\mu_1, \mu_2 > 0$ are imposed. This system naturally arises as a counterpart of the Brezis-Nirenberg problem (Comm. Pure Appl. Math. 36: 437-477, 1983).

Keywords:

Coupled nonlinear Schrödinger equations,
critical exponent,
Nehari manifold.

Mathematics Subject Classification: Primary: 35A15; Secondary: 35B33, 35B40, 35J50.

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