Papers › Constructive proofs for localized radial solutions of semilinear elliptic systems on ℝᵈ
Constructive proofs for localized radial solutions of semilinear elliptic systems on ℝᵈ
Jan Bouwe van den Berg, Olivier Hénot, Jean-Philippe Lessard
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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 ℝᵈ. 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 ℝ³, the Swift-Hohenberg equation on ℝ², and a three-component FitzHugh-Nagumo system on ℝ². These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.
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