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: R^d+k Rᵐ u: R d + k → R m of the system u (x) = W (u (x) ) Δ u (x) = ∇ W (u (x) ) (with W: Rᵐ R W: R m → R), corresponding to some nontrivial stable solutions e: Rᵏ Rᵐ e: R k → R m. The method we propose is based on a reduction to a ground state problem in a space of functions H H, where e is viewed as a local minimum of an effective potential defined in H H. As an application, by considering a heteroclinic orbit e: R Rᵐ e: R → R m, we obtain nontrivial solutions u: R^d+1 Rᵐ u: R d + 1 → R m (d 2 d ≥ 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.