Liouville type theorems for stable solutions of certain elliptic systems

TL;DR

This paper proves Liouville theorems for stable solutions of elliptic systems, showing non-existence under certain conditions, and extends results to bounded domains and weighted systems, with applications to classical models.

Contribution

It establishes new Liouville theorems for semi-stable solutions of elliptic systems, including cases with zero infimum of derivatives, and extends stability results to bounded domains and weighted systems.

Findings

No semi-stable solutions in certain dimensions for systems with positive derivative infimum.

Any positive semi-stable solution of the Lane-Emden system under specified conditions is constant.

Extension of stability results to bounded domains and weighted elliptic systems.

Abstract

We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any $C^2(\mathbb{R}^N)$ positive entire semi-stable solution of the following Lane-Emden system, {eqnarray*} \hbox{$(N_{\lambda,\gamma})$}50pt \{{array}{lcl} \hfill -\Delta u&=&\lambda f(x) \ v^p, \hfill -\Delta v&=&\gamma f(x) \ u^q, {array}.{eqnarray*} is necessarily constant, whenever the dimension $N< 8+3\alpha+\frac{8+4\alpha}{q-1}$, provided $p=1$, $q\ge2$ and $f(x)= (1+|x|^2)^{\frac{\alpha}{2}} $. The same also holds for $p=q\ge2$ provided $N < 2+ \frac{2(2+\alpha)}{p-1} (p+\sqrt{p(p-1)})$. We also consider the case of bounded domains $\Omega\subset\mathbb{R}^N$, where we extend results of Brown et al. \cite{bs} and Tertikas \cite{te} about stable solutions of equations to systems. At the end, we prove a Pohozaev type theorem for certain weighted elliptic systems.