On the Nonexistence of Entire Solutions to Nonlinear Second Order Elliptic Equations

Abstract

In this paper, it is shown that there are no global solutions (in all of $R^n $) to nonlinear elliptic equations of the form \[ \sum_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{{\partial u}}{{\partial x_i }}} \right)} = f(x,u,{\operatorname{grad}}u)\] if $f(x,u,p) \geqq g(u)$, where g is a convex, nonnegative function (nondecreasing if $n > 2$) such that $g^{ - {1 / 2}} $ is integrable near infinity unless $g(u(x)) \equiv 0$.