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We consider the following Gelfand type elliptic problem involving advection
where 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 for .On the other hand, combining Liapunov–Schmidt reduction method, we prove that it possesses a solution for . Besides, if is divergence free and satisfies a smallness condition, then the equation above admits a stable solution for .
Cite this article
Baishun Lai, Lulu Zhang, Gelfand Type Elliptic Problem Involving Advection. Z. Anal. Anwend. 36 (2017), no. 3 pp. 283–295