Liouville type theorems for stable solutions of the weighted elliptic system

TL;DR

This paper establishes Liouville type theorems for positive stable solutions of a weighted elliptic system in d6, identifying conditions on dimension, weight, and nonlinearity that guarantee nonexistence of solutions.

Contribution

It provides new Liouville theorems for stable solutions of a weighted elliptic system, extending known results to include weights and broader parameter ranges.

Findings

Nonexistence of solutions for dimensions d6 a0d7 12+5a0d5 for all p>1 and a0d7 a0 12+a0d5a0a0a0a0a0a0a0a0a0a0 a0d5a0a0a0a0a0a0a0a0a0a0 solutions for the weighted elliptic system.

The results depend on the dimension, weight parameter a0d6, and the nonlinearity exponent p, with explicit bounds provided.

Abstract

We examine the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=(1+|x|^2)^{\frac{\alpha}{2}} v,\\ -\Delta v=(1+|x|^2)^{\frac{\alpha}{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$, $p>1$ and $\alpha >0$. We prove Liouville type results for the classical positive (nonnegative) stable solutions in dimension $N<\ell+\dfrac{\alpha (\ell-2)}{2}$ ($N <\ell+\dfrac{\alpha (\ell-2)(p+3)}{4(p+1)}$) and $\ell \ge 5$, $p \in (1,p_*(\ell))$. In particular, for any $p>1$ and $\alpha > 0$, we obtain the nonexistence of classical positive (nonnegative) stable solutions for any $N \le 12+5 \alpha$ ($N\le 12+\dfrac{5\alpha (p+3)}{2(p+1)}$).