Boundary blow up for semilinear elliptic equations with nonlinear gradient terms

Abstract

The paper deals with the equation $\Delta u \pm |\nabla u|^q = f (u)$ in $\Omega \subset \mathbf{R}^n$, where $u$ blows up at the boundary $\partial \Omega$ and $\Omega$ is a bounded domain, which satisfies an interior and an exterior sphere condition. The existence and the asymptotic behaviour of $u$ near the boundary are investigated, showing how the nonlinear gradient term affects the results.