Nonexistence of Positive Supersolution to a Class of Semilinear Elliptic Equations and Systems in an Exterior Domain

Abstract

In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -\Delta u= h(x,u)\quad \ &{\rm in}\ \Omega,\\[1.5mm] \phantom{ -\Delta } \displaystyle u\ge 0\qquad &{\rm on}\ \partial{\Omega}, \end{array}\right. \end{eqnarray*} where $\Omega$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive supersolution to an elliptic system.