Liouville theorem and a priori estimates of radial solutions for a non-cooperative elliptic system. (September 2022)

• Record Type: Journal Article
• Authors: Quittner, Pavol
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: 35J10 -- 35J47 -- 35J61 -- 35B08 -- 35B45 -- 35B53 -- 35K58; Liouville theorem -- A priori estimate -- Elliptic system -- Schrödinger equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112971
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that p = 2 q + 3 > 1 is Sobolev subcritical, n ≤ 3 and β ∈ R . We first prove a Liouville theorem for the system − Δ u = | u | 2 q + 2 u + β | v | q + 2 | u | q u, − Δ v = | v | 2 q + 2 v + β | u | q + 2 | v | q v, i n R n, in the class of radial functions ( u, v ) such that the number of nodal domains of u, v, u − v, u + v is finite. Then we use this theorem to obtain a priori estimates of solutions to related elliptic systems. In the cubic case q = 0, those solutions correspond to the solitary waves of a system of Schrödinger equations, and their existence and multiplicity have been intensively studied by various methods. One of those methods is based on a priori estimates of suitable global solutions of corresponding parabolic systems. Unlike the previous studies, our Liouville theorem yields those estimates for all q ≥ 0 which are Sobolev subcritical.