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