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

Volume 109 / 2013

Li Chen, Yujuan Chen, Dang Luo 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 $p=2$.

Authors

  • 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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