We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions \(u:\mathbb {R}^{d+k}\rightarrow \mathbb {R}^m\) of the system \(\Delta u(x)=\nabla W(u(x))\) (with \(W:\mathbb {R}^m\rightarrow \mathbb {R}\) ), corresponding to some nontrivial stable solutions \(e:\mathbb {R}^k\rightarrow \mathbb {R}^m\) . The method we propose is based on a reduction to a ground state problem in a space of functions \(\mathcal {H}\) , where e is viewed as a local minimum of an effective potential defined in \(\mathcal {H}\) . As an application, by considering a heteroclinic orbit \(e:\mathbb {R}\rightarrow \mathbb {R}^m\) , we obtain nontrivial solutions \(u:\mathbb {R}^{d+1}\rightarrow \mathbb {R}^m\) ( \(d\ge 2\) ), converging asymptotically to e, which can be seen as the homoclinic analogs of the heteroclinic double layers, initially constructed by Alama-Bronsard-Gui and Schatzman.
