Gelfand type elliptic problem involving advection
type: Item
instance of: scholarly article
title: Gelfand type elliptic problem involving advection
zbMATH Open document ID: 1375.35166
DOI: 10.4171/ZAA/1589
author: Baishun Lai
author: Lu-Lu Zhang
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2017-08-16T00:00:00Z
review text: 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 \(N\geq 11\).
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35J61
zbMATH DE Number: 6761357
zbMATH Keywords: Gelfand type elliptic problem
zbMATH Keywords: stable solutions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/1589
OpenAlex ID: W2735482128
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description: scientific article; zbMATH DE number 6761357