Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 08, pp. 1-6.
Title: Simplicity and stability of the first eigenvalue
of a (p;q) Laplacian system
Authors: Ghasem A. Afrouzi (Univ. of Mazandaran, Babolsar, Iran)
Maryam Mirzapour (Univ. of Mazandaran, Babolsar, Iran)
Qihu Zhang (Zhengzhou Univ. of Light Industry, Henan, China)
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.