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Zbl 1169.35021
Garc{\'\i}a-Melián, Jorge
Large solutions for an elliptic system of quasilinear equations. (English)
[J] J. Differ. Equations 245, No. 12, 3735-3752 (2008). ISSN 0022-0396

Summary: We consider the quasilinear elliptic system $\Delta_pu= u^av^b$, $\Delta_pv= u^cv^e$ in a smooth bounded domain $\Omega\subset\Bbb R^N$, with the boundary conditions $u=v=+\infty$ on $\partial\Omega$. The operator $\Delta_p$ stands for the $p$-Laplacian defined by $\Delta_pu= \text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, and the exponents verify $a,e>p-1$, $b,c>0$ and $(a-p+1)(e-p+1)\ge bc$. We analyze positive solutions in both components, providing necessary and sufficient conditions for existence. We also prove uniqueness of positive solutions in the case $(a-p+1)(e-p+1)>bc$ and obtain the exact blow-up rate near the boundary of the solution. In the case $(a-p+1)(e-p+1)=bc$, infinitely many positive solutions are constructed.

MSC 2000:: *35J55 Systems of elliptic equations, boundary value problems
35J60 Nonlinear elliptic equations

Keywords: elliptic system of quasilinear equations; positive solutions; existence

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