Constructive proofs for localised radial solutions of semilinear elliptic systems on Rd

Abstract

Abstract Ground state solutions of elliptic problems have been analysed 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 localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on 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 localised radial solutions of the cubic Klein–Gordon equation on R 3 , the Swift–Hohenberg equation on R 2 , and a three-component FitzHugh–Nagumo system on R 2 . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.