JournalszaaVol. 36 , No. 3pp. 283–295

Gelfand Type Elliptic Problem Involving Advection

  • Baishun Lai

    Henan University, Kaifeng, China
  • Lulu Zhang

    Henan University, Kaifeng, China
Gelfand Type Elliptic Problem Involving Advection cover
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Abstract

We consider the following Gelfand type elliptic problem involving advection

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

where a(x)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)a(x) for N9N\leq 9.On the other hand, combining Liapunov–Schmidt reduction method, we prove that it possesses a solution for N4N\geq 4. Besides, if aa is divergence free and satisfies a smallness condition, then the equation above admits a stable solution for N11N\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