Gelfand Type Elliptic Problem Involving Advection

Abstract

We consider the following Gelfand type elliptic problem involving advection

where $a(x)$ is a smooth vector field. According to energy estimates, we obtain the nonexistence results of stable solution for this equation under some restrict conditions about $a(x)$ for $N\le 9$.On the other hand, combining Liapunov–Schmidt reduction method, we prove that it possesses a solution for $N\ge 4$. Besides, if $a$ is divergence free and satisfies a smallness condition, then the equation above admits a stable solution for $N\ge 11$.