Estimates for large solutions of a quasilinear elliptic equation

Abstract

Let $\Omega$ be a bounded smooth domain of $R^{n}$. We study the asymptotic behaviour of the solutions to the equation $\triangle u-|Du|^{q}=f(u)$ in $\Omega , 1<q<2,$ which satisfy the boundary condition $u(x)\rightarrow \infty$ as $x\rightarrow \partial \Omega$. These solutions are called large or blowup solutions. Near the boundary we give lower and upper bounds for the ratio $\psi (u)/\delta$, where $\psi (u) = \int _{u}^{\infty }1/\sqrt{2F}dt$, $F'=f$, $\delta =dist(x,\partial \Omega )$ or for the ratio $u/\delta ^{(2-q)/(1-q)}$. When in particular the ratio $f/F^{q/2}$is regular at infinity, we find again known results (Bandle and Giarrusso, in Adv Diff Equ 1, 133–150, 1996; Giarrusso, in Comptes Rendus de l’Acad Sci 331, 777–782 2000).