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Large solutions for an elliptic system of quasilinear equations. (English) Zbl 1169.35021

Summary: We consider the quasilinear elliptic system \(\Delta_pu= u^av^b\), \(\Delta_pv= u^cv^e\) in a smooth bounded domain \(\Omega\subset\mathbb R^N\), 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 \((a-p+1)(e-p+1)\geq bc\). 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:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations
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