Qualitative properties of H\'enon type equations with exponential nonlinearity

TL;DR

This paper investigates the qualitative behavior of solutions to Hénon type equations with exponential nonlinearities, including classification of stable solutions and asymptotic analysis of radial solutions across various dimensions.

Contribution

It provides a complete classification of stable solutions at infinity and detailed asymptotic behavior of radial solutions for these equations.

Findings

Classification of stable solutions at infinity in  and higher dimensions.

Existence and asymptotic behavior of entire radial solutions.

Complete classification of stable and stable at infinity solutions in any dimension.

Abstract

We are interested in the qualitative properties of solutions of the H\'enon type equations with exponential nonlinearity. First, we classify the stable at infinity solutions of $\Delta u +|x|^\alpha e^u=0$ in $\mathbb{R}^N$, which gives a complete answer to the problem considered in [Wang-Ye]. Secondly, existence and precise asymptotic behaviors of entire radial solutions to $\Delta^2 u=|x|^{\alpha} e^u$ are obtained. Then we classify the stable and stable at infinity radial solutions to $\Delta^2 u=|x|^{\alpha} e^u$ in any dimension.