January  2008, 7(1): 125-141. doi: 10.3934/cpaa.2008.7.125

Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains

1. 

Università "La Sapienza" Roma I, Dipartimento di Matematica, Piazzale Aldo Moro 2, I-00185 Roma, Italy

Received  August 2006 Revised  February 2007 Published  October 2007

In this paper we consider fully nonlinear elliptic operators of the form $F(x,u,Du,D^2u)$. Our aim is to prove that, under suitable assumptions on $F$, the only nonnegative viscosity super-solution $u$ of $F(x,u,Du,D^2u)=0$ in an unbounded domain $\Omega$ is $u\equiv 0$. We show that this uniqueness result holds for the class of nonnegative super-solutions $u$ satisfying

limin$f_{x\in\Omega, |x|\to\infty}\frac{u(x)+1}{\dist(x,\partial\Omega)}=0,$

and then, in particular, for strictly sublinear super-solutions in a domain $\Omega$ containing an open cone. In the special case that $\Omega=\mathbb R^N$, or that $F$ is the Bellman operator, we show that the same result holds for the whole class of nonnegative super-solutions.
Our principal assumption on the operator $F$ involves its zero and first order dependence when $|x|\to\infty$. The same kind of assumption was introduced in a recent paper in collaboration with H. Berestycki and F. Hamel [4] to establish a Liouville type result for semilinear equations. The strategy we follow to prove our main results is the same as in [4], even if here we consider fully nonlinear operators, possibly unbounded solutions and more general domains.

Citation: Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure and Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125
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