In this paper, we study positive
solution of the following system of quasilinear elliptic equations
div$(|\nabla u|^{p-2}\nabla u)=u^{m_1}v^{n_1},$ in $\Omega$
div$(|\nabla v|^{q-2}\nabla v)=u^{m_2}v^{n_2},$ in $\Omega,$ $\qquad\qquad\qquad\qquad$ (0.1)
where
$m_1>p-1,n_2>q-1, m_2,n_1>0$, and $\Omega\subset R^N$ is a smooth
bounded domain, subject to three different types of Dirichlet
boundary conditions: $u=\lambda, v=\mu$ or $u=v=+\infty$ or
$u=+\infty, v=\mu$ on $\partial\Omega$, where $\lambda, \mu>0$.
Under several hypotheses on the parameters $m_1,n_1,m_2,n_2$, we
show that the existence
of positive solutions. We further provide the asymptotic
behaviors of the solutions near $\partial\Omega$. Some more general
related problems are also studied.
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