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