A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems

Xianjin Chen; Jianxin Zhou

doi:10.3934/proc.2009.2009.151

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2009, Volume 2009: 151-160. Doi: 10.3934/proc.2009.2009.151 open access

This issue Previous Article On the minimal time null controllability of the heat equation Next Article Stability of linear dynamic equations on time scales

A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems

Xianjin Chen^1, and
Jianxin Zhou^2,

1.
Department of Mathematics, Texas A&M University, College Station, TX 77843
2.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Received: June 30, 2008

Revised: April 30, 2009

Published: August 2009

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Abstract

The aim of this paper is to numerically investigate multiple solutions of semilinear elliptic systems with zero Dirichlet boundary conditions

-$\Delta u=F_u(x;u,v),$ $x\in\Omega,
-$\Delta v=F_v(x;u,v),$ $x\in\Omega,

where $\Omega \subset \mathbb{R}^{N}$ ($N\ge 1$) is a bounded domain. A strongly coupled case where the potential $F(x;u,v)$ takes the form $|u|^{\alpha_1}|v|^{\alpha_2}$ with $\alpha_1, \alpha_2>1$ is specially studied. By using a local min-orthogonal method, both positive and sign-changing solutions are found and displayed.

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Mathematics Subject Classification: Primary: 35A40, 35A15; Secondary: 58E05.

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