Monotonicity formulas for coupled elliptic gradient systems with applications

Advances in Nonlinear Analysis

article

Mostafa Fazly¹  Henrik Shahgholian¹ 

[1] Department of Mathematics, The University of Texas at San Antonio

关键词: Coupled elliptic systems;    monotonicity formulae;    homogeneous solutions;    stable solutions;    fractional Laplacian operator;   

DOI  :  10.1515/anona-2020-0010

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

Consider the following coupled elliptic system of equations (− Δ )sui=(u12+⋯ +um2)p− 12uiin  Rn, $$\begin{array}{} \displaystyle (-{\it\Delta})^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} ~~ \mathbb{R}^n , \end{array}$$ where 0 1, m ≥ 1, u = (ui)i=1m $\begin{array}{} \displaystyle (u_i)_{i=1}^m \end{array}$ and u i : ℝ n → 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 [26] and later Caffarelli, Gidas and Spruck in [6] 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 ≥ 1 and s = 1 a similar classification result is given by Druet, Hebey and Vétois in [17] 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 [1] 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.

【 授权许可】

CC BY