A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms

Abstract

We prove that if the elliptic problem $-\Delta u+b(x)|\nabla u|=c(x)u$ with $c\ge 0$ has a positive supersolution in a domain $\varOmega$ of ${\mathrm {R}}^{N\ge 3}$, then c, b must satisfy the inequality

$$\begin{aligned} \sqrt{ \int _\varOmega c\phi ^2}\le \sqrt{ \int _\varOmega | \nabla \phi |^2}+\sqrt{ \int _\varOmega \frac{b^2}{4}\phi ^2},\quad \phi \in C_c^\infty (\varOmega ). \end{aligned}$$

As an application, we obtain Liouville-type theorems for positive supersolutions in exterior domains when $c(x)-\frac{b^2(x)}{4}>0$ for large |x|, but unlike the known results, we allow the case $\lim _{|x|\rightarrow \infty }c(x)-\frac{b^2(x)}{4}=0$. The weights b and c are allowed to be unbounded. In particular, among other things, we show that if $\tau :=\limsup _{|x| \rightarrow \infty }|xb(x)|<\infty,$ then this problem does not admit any positive supersolution if

$$\begin{aligned} \liminf _{|x| \rightarrow \infty }|x|^2c(x)> \frac{(N-2+\tau )^2}{4}, \end{aligned}$$

and, when $\tau =\infty ,$ we have the same if

$$\begin{aligned} \limsup _{R\rightarrow \infty } R\left( \frac{ \inf _{R<|x|<2 R} (c(x)-\frac{b(x)^2}{4})}{\sup _{\frac{R}{2}<|x|<4 R}|b(x)|}\right) =\infty . \end{aligned}$$