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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 36, Issue 3, 2017, pp. 283–295**

**DOI: 10.4171/ZAA/1589**

Published online: 2017-07-17

Gelfand Type Elliptic Problem Involving Advection

Baishun Lai^{[1]}and Lulu Zhang

^{[2]}(1) Henan University, Kaifeng, China

(2) Henan University, Kaifeng, China

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 $N\geq11$.

*Keywords: *Gelfand problem, stability, nonexistence, advection

Lai Baishun, Zhang Lulu: Gelfand Type Elliptic Problem Involving Advection. *Z. Anal. Anwend.* 36 (2017), 283-295. doi: 10.4171/ZAA/1589