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

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:Rᵈ⁺ᵏ→ Rᵐ u : R d + k → R m of the system Δ 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:Rᵈ⁺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.