Boundary blow-up solutions: gradient asymptotics and uniqueness

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $\Delta u=f(u)$ in $\Omega$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open even for smooth convex domains. We identify the normalized gradient $Q_u=|\nabla u{|}^2/(2F(u))$, $F^{\prime }=f$, as a quantity governing uniqueness. Under a structural condition on $f$, a boundary blow-up solution $u$ is unique if ${\mathrm{limsup}}_{x\to \partial \Omega }{Q}_u(x)\le 1$, without any regularity assumption on $\partial \Omega$. For $C^{1,1}$ domains, assuming a growth condition on $f$, we prove $Q_u(x)\to 1$ for every boundary blow-up solution and hence obtain uniqueness under the structural condition. For convex domains, we prove $Q_u\le 1$ for the minimal boundary blow-up solution and obtain uniqueness when $\sqrt{F}$ is eventually convex, without imposing any additional boundary regularity.

26 pages