Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case

Abstract

In this paper, we study positivesolution 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 smoothbounded domain, subject to three different types of Dirichletboundary 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 $m_1,n_1,m_2,n_2$, weshow that the existence of positive solutions. We further provide the asymptoticbehaviors of the solutions near $\partial\Omega$. Some more generalrelated problems are also studied.