On the elliptic problem Δu-|∇u| q +λu p =0

Abstract

We consider regular solutions of the elliptic problem (1) $\Delta u-|\nabla u|^q+\lambda u^p=0$, and $u>0$ for $x\in\Omega $, $u=0$ on $x\in\partial \Omega $, where $\Omega \subset \bold R^n$ is a smooth, bounded domain, $u\colon \overline{\Omega }\rightarrow \bold R$ is $C^2$, and $q, p, \lambda$ are parameters satisfying $q, p>1$ and $\lambda >0$. The motivation for studying problem (1) comes from a related parabolic problem, $(2)\ v_t=\Delta v-|\nabla v|^q+v^p\ (t>0,\ x\in\Omega )$, $v(0, x)=\varphi(x)\ge 0\ (x\in\Omega )$, $v(t, x)=0\ (t>0$, $x\in\partial \Omega )$, where now $v=v(t, x)$. Previously, we investigated whether there exist initial values $\varphi$ for which the resulting solution of (2) blows up in finite time. It turns out that a natural candidate for such a $\varphi$ is a solution of (1). More precisely, assume either $(3)\ 1<q< 2p/(p+1)$, $n/2<(p+1)/(p-1)$, $\lambda>0$ is sufficiently small, or $(4)\ q=2p/(p+1)$, $p$ is sufficiently large, $0<\lambda\le 2/(p+1)$. It follows that if $\varphi$ is a solution of (1), then the resulting solution of (2) blows up in finite time.
Thus we are led to investigate under what conditions there exist solutions of (1) and whether these solutions are unique.