On stable entire solutions of semi-linear elliptic equations with weights

TL;DR

This paper investigates the existence and non-existence of stable solutions to weighted semi-linear elliptic equations in Euclidean space, focusing on how weights and dimension influence solution stability.

Contribution

It provides new non-existence results for stable solutions depending on weights, dimension, and nonlinearity, extending previous methods with optimal conditions.

Findings

Non-existence results depend on dimension, weights, and nonlinearity.

Monotonicity of weights affects solution stability.

Optimal non-existence conditions are established for specific weight classes.

Abstract

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of {equation} \label{pop} -div(\omega_1 \nabla u) = \omega_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth weights $ \omega_1(x),\omega_2(x)$. We consider the cases $ f(u) = e^u, u^p$ where $p>1$ and $ -u^{-p}$ where $ p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $ p$ and the behaviour of $ \omega_1,\omega_2$ near infinity. Also the monotonicity of $ \omega_1$ is involved in some results. Our methods here are the methods developed by Farina, \cite{f2}. We examine a specific class of weights $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x) = (|x|^2+1)^\frac{\beta}{2} g(x)$ where $ g(x)$ is a positive function with a finite limit at $ \infty$. For this class of weights non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.