Qualitative properties of stable solutions to some supercritical problems

Abstract

In this paper, we study symmetry properties of stable solutions to the Lane-Emden equation $ \Delta u+|u|^{p-1}u = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $ with $ n\geq 11 $, $ p $ in a suitable range and the Liouville equation $ \Delta u+e^{u} = 0\quad{\rm{in}}\quad\mathbb{R}^{n} $ with $ n = 10 $.

Keywords

• lane-emden equation
• liouville equation
• stable solution
• rigidity results
• symmetry