Monotonicity formulas for coupled elliptic gradient systems with applications

TL;DR

This paper develops monotonicity formulas for coupled elliptic systems with local and nonlocal operators, enabling classification of solutions and addressing challenges in extending scalar results to systems.

Contribution

It introduces new monotonicity formulas for coupled elliptic systems with fractional and integer order derivatives, advancing understanding of solution behavior.

Findings

Derived monotonicity formulas for local and nonlocal systems

Classified finite Morse index solutions using these formulas

Highlighted open problems for Lane-Emden systems

Abstract

Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-\Delta)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later Caffarelli, Gidas and Spruck in \cite{cgs} provided the classification of solutions for Sobolev sub-critical and critical exponents. More recently, for the case of local system of equations that is when $m\ge1$ and $s=1$ a similar classification result is given by Druet, Hebey and V\'{e}tois in \cite{dhv} and references therein. In this paper, we derive monotonicity formulae for entire solutions of the above local, when $s=1,2$, and nonlocal, when $0<s<1$ and $1<s<2$, system. These monotonicity formulae are of great interests due to the fact that a counterpart of the celebrated monotonicity formula of Alt-Caffarelli-Friedman \cite{acf} seems to be challenging to derive for system of equations. Then, we apply these formulae to give a classification of finite Morse index solutions. In the end, we provide an open problem in regards to monotonicity formulae for Lane-Emden systems.