Boundary estimates for solutions of weighted semilinear elliptic equations

Let $b(x)$ be a positive function in a bounded smooth domain $\Omega\subset R^N$, and let $f(t)$ be a positive non decreasing function on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$. We investigate boundary blow-up solutions of the equation $\Delta u=b(x)f(u)$. Under appropriate conditions on $b(x)$ as $x$ approaches $\partial\Omega$ and on $f(t)$ as $t$ goes to infinity, we find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.   We also investigate positive solutions of the equation $\Delta u+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on $\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$ denotes the distance from $x$ to $\partial\Omega$. We find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.