Liouville type theorems for two elliptic equations with advections
Volume 122 / 2019
Annales Polonici Mathematici 122 (2019), 11-20
MSC: Primary 35B53, 35J60; Secondary 35B35.
DOI: 10.4064/ap180312-20-9
Published online: 11 January 2019
Abstract
We study the elliptic equations $$ -\Delta u+a(x)\cdot\nabla u=f(u)\ \quad\mbox{in }\mathbb R^{N}, $$ where $ N\geq 3 $, the advection term $a(x)$ is a smooth vector field satisfying a certain decay condition and the nonlinearity $f(u)$ is of the form $-u^{-p},\;p \gt 0,$ or $e^u$. We establish Liouville type theorems for the class of positive stable solutions when $f(u)=-u^{-p}$ and for the class of stable solutions when $f(u)=e^u$. In particular, our results improve some results of B. Lai and L. Zhang [Z. Anal. Anwend. 36 (2017), 283–295] and of L. Ma and J. C. Wei [J. Funct. Anal. 254 (2008), 1058–1087].
