Gelfand type elliptic problem involving advection
Summary: We consider the following Gelfand type elliptic problem involving advection \[ -\Delta u+a(x) \cdot \nabla u=e^{u}\quad \text{in}\,\, \mathbb R^{N}\text{,} \] 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 \(N\geq 11\).
scientific article; zbMATH DE number 6761357