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On positive solutions for a class of singular quasilinear elliptic systems. (English) Zbl 1155.35024

The authors deal with the existence of positive weak solution to the quasilinear elliptic system with weights, that is \[ \begin{cases} -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)= \lambda|x|^{-(a+ 1)p+ c_1} u^\alpha\vee v^\gamma\;&\text{in }\Omega,\\ -\text{div}(|x|^{-bq}|\nabla u|^{p-2}\nabla u)= \lambda|x|^{-(b+1)q+ c_2}\;&\text{in }\Omega,\\ u= v= 0\;&\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a bounded smooth domain of \(\mathbb{R}^d\), \(1< p,\,q< d\). Using the lower and upper solution method, the authors under suitable assumptions on the data of (1) prove the existence of positive weak solution of (1). Moreover, the authors present a nonexistence result for (1) as well.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations
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