A nonexistence result of stable solutions to a weighted elliptic equation

Abstract

We investigate the following weighted elliptic equation
$$
-\Delta u+b(x)\cdot\nabla u=(1+|x|^2)^{ \frac{\alpha} {2}}u^{p} \mbox{ in }\mathbb R^N,
$$
where $\alpha\geq 0$, $p>1$, $N \geq 3$ and the advection term $b(x)$ is a smooth vector field satisfying certain decay condition. We establish a Liouville-type theorem for possible stable solutions of the equation above.