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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:
\[ \begin{aligned} \text{div}(| \nabla u| ^{p-2}\nabla u)&= u^{m_{1}}v^{n_{1}} \quad\text{in }\Omega,\\ \text{div}(| \nabla v| ^{q-2}\nabla v)&= u^{m_{2}}v^{n_{2}} \quad\text{in }\Omega, \end{aligned} \]
where \(m_{1}>p-1\), \(n_{2}>q-1\), \(m_{2}, n_{1}>0\), and \(\Omega\subset \mathbb R^{N}\) 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 to PDEs
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