A priori estimates, existence and non-existence for quasilinear cooperative elliptic systems

H. Zou

Sbornik. Mathematics, Tome 199 (2008) no. 4, pp. 557-578

Résumé

Let $m>1$ be a real number and let $\Omega\subset\mathbb R^n$, $n\geqslant2$, be a connected smooth domain. Consider the system of quasi-linear elliptic differential equations \begin{align*} \operatorname{div}(|\nabla u|^{m-2}\nabla u)+f(u,v)=0\quad\text{in } \Omega, \\ \operatorname{div}(|\nabla v|^{m-2}\nabla v)+g(u,v)=0\quad\text{in } \Omega, \end{align*} where $u\geqslant0$, $v\geqslant0$, $f$ and $g$ are real functions. Relations between the Liouville non-existence and a priori estimates and existence on bounded domains are studied. Under appropriate conditions, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established. Bibliography: 11 titles.