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 cb 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

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

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Acknowledgements

The authors thank the referees for their valuable suggestions to improve the presentation of the original manuscript. A. Aghajani was partially supported by Grant from IPM (No. 99350212). C. Cowan supported in part by NSERC

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Aghajani, A., Cowan, C. A note on the nonexistence of positive supersolutions to elliptic equations with gradient terms. Annali di Matematica 200, 125–135 (2021). https://doi.org/10.1007/s10231-020-00987-2

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Keywords

  • Liouville-type theorems
  • Supersolutions
  • Gradient term

Mathematics Subject Classification

  • 35J60
  • 35B53