Liouville theorem and a priori estimates of radial solutions for a non-cooperative elliptic system

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=2q+3>1 is Sobolev subcritical, n≤3 and β∈R. We first prove a Liouville theorem for the system −Δu=|u|2q+2u+β|v|q+2|u|qu,−Δv=|v|2q+2v+β|u|q+2|v|qv,inRn, 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.