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.


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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