Electron. J. Diff. Equ., Vol. 2012 (2012), No. 08, pp. 1-6.

Simplicity and stability of the first eigenvalue of a (p;q) Laplacian system

Ghasem A. Afrouzi, Maryam Mirzapour, Qihu Zhang

Abstract:
This article concerns special properties of the principal eigenvalue of a nonlinear elliptic system with Dirichlet boundary conditions. In particular, we show the simplicity of the first eigenvalue of
$$\displaylines{
 -\Delta_p u = \lambda |u|^{\alpha-1}|v|^{\beta-1}v \quad
 \hbox{in } \Omega,\cr
 -\Delta_q v = \lambda |u|^{\alpha-1}|v|^{\beta-1}u
 \quad \hbox{in } \Omega,\cr
 (u,v)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega),
 }$$
with respect to the exponents p and q, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$.

Submitted November 3, 2011. Published January 12, 2012.
Math Subject Classifications: 35J60, 35B30, 35B40.
Key Words: Eigenvalue problem; quasilinear operator; simplicity; stability.

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Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email: afrouzi@umz.ac.ir
Maryam Mirzapour
Department of Mathematics, Faculty of Mathematical Sciences
University of Mazandaran, Babolsar, Iran
email mirzapour@stu.umz.ac.ir
  Qihu Zhang
Department of Mathematics and Information Science
Zhengzhou University of Light Industry
Zhengzhou, Henan 450002, China
email: zhangqh1999@yahoo.com.cn

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