Ground state of some variational problems in Hilbert spaces and applications to P.D.E

Abstract

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 $u:{\mathbb{R}}^{d+k}\to {\mathbb{R}}^m$ of the system $\Delta u(x)=\nabla W(u(x))$ (with $W:{\mathbb{R}}^m\to \mathbb{R}$), corresponding to some nontrivial stable solutions $e:{\mathbb{R}}^k\to {\mathbb{R}}^m$. 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 $e:\mathbb{R}\to {\mathbb{R}}^m$, we obtain nontrivial solutions $u:{\mathbb{R}}^{d+1}\to {\mathbb{R}}^m$ ($d\ge 2$), 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.