Monotonicity formulas for coupled elliptic gradient systems with applications

Abstract

Consider the following coupled elliptic system of equations (−∆) s u i = (u 2 1 + · · · + u 2 m )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.

On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand–Liouville equation

“…The key technique of our proof is a monotonicity formula that is developed in this section. The ideas and methods in this section are motivated by the ones in [20], see also [13,19,33], and references therein. Define E(r, x 0 , u e ) := r 2s−n…”

Stable and finite Morse index solutions of a nonlinear Schrödinger system

“…where C 0 = 0 if u is a stable solution. Notice that (10) follows immediately from (15). Choosing m = p+1 p−1 > 1 and applying Young's inequality, we obtain…”

“…This new method was successfully exploited to deal with other types of elliptic problems, see [11][12][13]15,16,27,32] and the references therein.…”

On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation

“…The classification of stable and finite Morse index solutions using monotonicity formulas was initiated in a series of articles [10,11,34] for the Lane-Emden equation and [36] for the Liouville type equations. We refer interested readers to [18] for an introduction regarding monotonicity formula in this context. Compared to the Lane-Emden equation, proving a priori estimates for the weighted "Dirichlet" energy and for the boundary integral of linear term are more challenging when dealing with the Gelfand-Liouville equation.…”