  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.
