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Boundary blowup for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights
Existence of boundary blowup solutions for a class of quasiliner elliptic systems for the subcritical case
1.  Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Jiangsu Nanjing 210097, China, China 
div$(\nabla u^{p2}\nabla u)=u^{m_1}v^{n_1},$ in $\Omega$
div$(\nabla v^{q2}\nabla v)=u^{m_2}v^{n_2},$ in $\Omega,$ $\qquad\qquad\qquad\qquad$ (0.1)
where $m_1>p1,n_2>q1, m_2,n_1>0$, 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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