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## Boundary blow-up solutions for a cooperative system involving the $p$-Laplacian

### Tom 109 / 2013

Annales Polonici Mathematici 109 (2013), 297-310 MSC: Primary 35J57; Secondary 35B40. DOI: 10.4064/ap109-3-5

#### Streszczenie

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 $p=2$.

#### Autorzy

• Li ChenDepartment of Mathematics
Nantong University
226007, Nantong, P.R. China
• Yujuan ChenDepartment of Mathematics
Nantong University
226007, Nantong, P.R. China
e-mail
• Dang LuoCollege of Mathematics and
Information Science
North China University of
Water Resources and Electric Power
450011, Zhengzhou, P.R. China

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