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

We prove that if the elliptic problem $-\mathrm{\Delta }u+b(x)\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=c(x)u$ with $c\ge 0$ has a positive supersolution in a domain $\mathrm{\Omega }$ of ${\text{IR}}^{N\ge 3}$, then $c,b$ must satisfy the inequality $$\sqrt{{\int }_{\mathrm{\Omega }}c{\varphi }^2}\le \sqrt{{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }\varphi {\mathrm{\mid }}^2}+\sqrt{{\int }_{\mathrm{\Omega }}\frac{b^2}{4}{\varphi }^2},\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$$ 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 $\mathrm{\mid }x\mathrm{\mid }$, but unlike the known results we allow the case ${\mathrm{lim\hspace{0.17em}inf}}_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}c(x)-\frac{b^2(x)}{4}=0$. Also the weights $b$ and $c$ are allowed to be unbounded. In particular, among other things, we show that if $\tau :={\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}\mathrm{\mid }xb(x)\mathrm{\mid }<\mathrm{\infty }$ then this problem does not admit any positive supersolution if $$\underset{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\mathrm{\mid }x{\mathrm{\mid }}^{2}c(x)>\frac{(N-2+\tau {)}^2}{4},$$ and, when $\tau =\mathrm{\infty },$ we have the same if $$\underset{R\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}R\left(\frac{\underset{R<\mathrm{\mid }x\mathrm{\mid }<2R}{\inf }(c(x)-\frac{b(x{)}^2}{4})}{\underset{\frac{R}{2}<\mathrm{\mid }x\mathrm{\mid }<4R}{\sup }\mathrm{\mid }b(x)\mathrm{\mid }}\right)=\mathrm{\infty }\mathrm{.}$$