Boundary blow-up solutions for a cooperative system involving the $p$-Laplacian
Volume 109 / 2013
Annales Polonici Mathematici 109 (2013), 297-310
MSC: Primary 35J57; Secondary 35B40.
DOI: 10.4064/ap109-3-5
Abstract
We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system $\varDelta _p u=g(u-\alpha v),$ $\varDelta _p v=f(v-\beta u)$ in a smooth bounded domain of $\mathbb {R}^N$, where $\varDelta _p$ is the $p$-Laplacian operator defined by $\varDelta _p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$ with $p >1$, $f$ and $g$ are nondecreasing, nonnegative $C^1$ functions, and $\alpha $ and $\beta $ are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for $p=2$.
