Constructive proofs for localized radial solutions of semilinear elliptic systems on $\mathbb{R}^d$

TL;DR

This paper introduces a constructive method combining dynamical systems theory and computer-assisted proofs to establish the existence of localized radial solutions in nonlinear elliptic systems on Euclidean spaces.

Contribution

It develops a general approach to prove localized solutions constructively, applicable to various elliptic systems, using Lyapunov-Perron operators and Newton-Kantorovich theorems.

Findings

Proved localized solutions for the cubic Klein-Gordon equation in R^3.

Established solutions for the Swift-Hohenberg equation in R^2.

Demonstrated solutions for a FitzHugh-Nagumo system in R^2.

Abstract

Ground state solutions of elliptic problems have been analyzed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localized solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localized radially symmetric solutions of elliptic systems on $\mathbb{R}^d$. Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov-Perron operator with a Newton-Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localized radial solutions of the cubic Klein-Gordon equation on $\mathbb{R}^3$, the Swift-Hohenberg equation on $\mathbb{R}^2$, and a three-component FitzHugh-Nagumo system on $\mathbb{R}^2$. These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.