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

作者：Rossi Luca^*

来源：Communications on Pure and Applied Analysis, 2008, 7(1): 125-141.

DOI：10.3934/cpaa.2008.7.125

• fully nonlinear elliptic equations
• positive solutions
• unbounded domains
• viscosity solutions

In this paper we consider fully nonlinear elliptic operators of the form F(x, u, Du, D(2)u). Our aim is to prove that, under suitable assumptions on F, the only nonnegative viscosity super-solution u of F(x, u, Du, D(2)u) = 0 in an unbounded domain Omega is u = 0. We show that this uniqueness result holds for the class of nonnegative super-solutions u satisfying <br xmlns:set="http://exslt.org/sets">lim inf(x is an element of Omega vertical bar x vertical bar -> infinity) u(x) + 1/dist(x, partial derivative 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 = 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 vertical bar x vertical bar -> infinity. The same kind of assumption was introduced in a recent paper in collaboration with H. Berestycki and F. Hamel [ ] to establish a Liouville type result for semilinear equations. The strategy we follow to prove our main results is the same as in [ ], even if here we consider fully nonlinear operators, possibly unbounded solutions and more general domains.

• 出版日期2008-1