Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights

Abstract

We consider the elliptic problems $\Delta u=a(x)u^m$, $m>1$, and$\Delta u=a(x)e^u$ in a smooth bounded domain $\Omega$, with theboundary condition $u=+\infty$ on $\partial\Omega$. The weightfunction $a(x)$ is assumed to be Hölder continuous, growing likea negative power of $d(x)=$ dist $(x,\partial\Omega)$ near$\partial\Omega$. We show existence and nonexistence results,uniqueness and asymptotic estimates near the boundary for both thesolutions and their normal derivatives.