Bifurcation results on positive solutions of an indefinite nonlinearelliptic system

Rushun Tian; Zhi-Qiang Wang

doi:10.3934/dcds.2013.33.335

Article Contents

2013, Volume 33, Issue 1: 335-344. Doi: 10.3934/dcds.2013.33.335

This issue Previous Article Infinitely many solutions for some singular elliptic problems Next Article Instability of periodic minimals

Bifurcation results on positive solutions of an indefinite nonlinear elliptic system

Rushun Tian^1, and
Zhi-Qiang Wang^2,

1.
Department of Mathematics & Statistics, Utah State University, Logan, UT 84322
2.
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322

Received: May 2011

Revised: October 2011

Published: January 2013

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Abstract

Consider the following nonlinear elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u - u=\mu_1u^3+\beta uv^2,\ & \hbox{in}\ \Omega\\ -\Delta v - v= \mu_2v^3+\beta vu^2,\ & \hbox{in}\ \Omega\\ u,v>0\ \hbox{in}\ \Omega, \ u=v=0,\ & \hbox{on}\ \partial\Omega, \end{array} \right. \end{equation*}where $\mu_1,\mu_2>0$ are constants and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\leq3$. We study the existence and non-existence of positive solutions and give bifurcation results in terms of the coupling constant $\beta$.

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Mathematics Subject Classification: Primary: 35B05, 35J61, 58C40; Secondary: 35J15, 58E07.

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