Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

Abstract

This paper is concerned withexact boundary behavior of large solutions to semilinear elliptic equations$\Delta~u~=b(x)f(u)(C_0+|\nabla~u|^q)$, $x\in~\xO$, where$\Omega$ is a bounded domain with a smooth boundaryin $\mathbb~R^N$,$C_0\geq~0$, $q\in~[0,~2)$, $b~\in~C^{\xa}_{\rm~loc}({\xO})$ ispositive in $\Omega$, but may be vanishing or appropriatelysingular on the boundary, $f\in~C[0,~\infty)$, $f(0)=0$, and $f$ isstrictly increasing on $[0,~\infty)$ (or $f\in~C(\mathbb~R)$,$f(s)>0$, $\forall~\,~s\in~\mathbb~R,$ $f$ is strictly increasing on$\mathbb~R$). We show unified boundary behavior of such solutions to the problem under a new structure condition on $f$.