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Existence of boundary blow-up solutions for a class of quasilinear elliptic systems for the subcritical case. (English) Zbl 1154.35028
The authors study positive solution of the following system of quasilinear elliptic equations: \aligned \text{div}(\vert \nabla u\vert ^{p-2}\nabla u)&= u^{m_{1}}v^{n_{1}} \quad\text{in }\Omega,\\ \text{div}(\vert \nabla v\vert ^{q-2}\nabla v)&= u^{m_{2}}v^{n_{2}} \quad\text{in }\Omega, \endaligned where $m_{1}>p-1$, $n_{2}>q-1$, $m_{2}, n_{1}>0$, and $\Omega\subset \Bbb R^{N}$ is a smooth bounded domain, subject to three different types of Dirichlet boundary conditions: $u=\lambda$, $v=\mu$, $u=v=+\infty$ or $u=+\infty$, $v=u$ on $\partial\Omega$, where $\lambda,\mu>0$. Under several hypotheses on the parameters $m_{1}, n_{1}, m_{2}, n_{2}$, the authors show the existence of positive solutions and provide the asymptotic hehavior of the solutions near $\partial\Omega$. Some more general related problems are also studied.

MSC:
 35J45 Systems of elliptic equations, general (MSC2000) 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35J60 Nonlinear elliptic equations 35B40 Asymptotic behavior of solutions of PDE
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