On positive solutions of quasilinear elliptic systems

Cheng, Yuanji

Czechoslovak Mathematical Journal, Tome 47 (1997) no. 4, pp. 681-687.

Résumé

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems \[ \left\rbrace \begin{array}{ll}-\Delta _p u = f(x,u,v), \quad \text{in} \ \Omega , -\Delta _p v = g(x,u,v), \quad \text{in} \ \Omega , u = v = 0, \quad \text{on} \ \partial \Omega , \end{array}\right.\] where $-\Delta _p$ is the $p$-Laplace operator, $p>1$ and $\Omega $ is a $C^{1,\alpha }$-domain in $\mathbb R^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)=\lambda _1 v^{p-1}$, $ g(x,u,v)=\lambda _2u^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.

Détail

Classification : 35B05, 35J55, 35J65, 35J70
Keywords: Eigenvalue problem; Degenerate elliptic operator; Nonlinear systems; Positive solutions.