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

TL;DR

This paper establishes conditions under which positive supersolutions to certain elliptic equations with gradient terms cannot exist, extending Liouville theorems to cases with unbounded weights and borderline behaviors.

Contribution

It provides new nonexistence criteria for positive supersolutions of elliptic equations with gradient terms, including cases with unbounded coefficients and near-critical growth conditions.

Findings

Derived inequality relating coefficients c and b for supersolutions.

Proved nonexistence of positive supersolutions under specific growth conditions.

Extended Liouville theorems to include borderline and unbounded coefficient cases.

Abstract

We prove that if the elliptic problem $-\Delta u+b(x)|\nabla u|=c(x)u$ with $c\ge0$ has a positive supersolution in a domain $\Omega$ of $ \IR^{N\ge 3}$, then $c,b$ must satisfy the inequality \[\sqrt{ \int_\Omega c\phi^2}\le \sqrt{ \int_\Omega | \nabla\phi|^2}+\sqrt{ \int_\Omega \frac{b^2}{4}\phi^2},~~~\phi \in C_c^\infty(\Omega).\] 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 $\liminf_{|x|\rightarrow\infty}c(x)-\frac{b^2(x)}{4}=0$. Also the weights $b$ and $c$ are allowed to be unbounded. In particular, among other things, we show that if $\tau:=\limsup_{|x| \rightarrow\infty}|xb(x)|<\infty$ then this problem does not admit any positive supersolution if \[\liminf_{|x| \rightarrow\infty}|x|^2c(x)> \frac{(N-2+\tau)^2}{4},\] and, when $\tau=\infty, $ we have the same if \[\limsup_{R\rightarrow\infty} R\Big(\frac{ \inf_{R<|x|<2 R} (c(x)-\frac{b(x)^2}{4})}{\sup_{\frac{R}{2}<|x|<4 R}|b(x)|}\Big)=\infty.\]