# Gelfand Type Elliptic Problem Involving Advection

### Baishun Lai

Henan University, Kaifeng, China### Lulu Zhang

Henan University, Kaifeng, China

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## Abstract

We consider the following Gelfand type elliptic problem involving advection

$-\Delta u+a(x) \cdot \nabla u=e^{u}\ \ \mbox{in}\ \mathbb R^{N},$

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 $N\leq 9$.On the other hand, combining Liapunov–Schmidt reduction method, we prove that it possesses a solution for $N\geq 4$. Besides, if $a$ is divergence free and satisfies a smallness condition, then the equation above admits a stable solution for $N\geq11$.

## Cite this article

Baishun Lai, Lulu Zhang, Gelfand Type Elliptic Problem Involving Advection. Z. Anal. Anwend. 36 (2017), no. 3 pp. 283–295

DOI 10.4171/ZAA/1589