Ground state of some variational problems in Hilbert spaces and applications to P.D.E
Ioannis Arkoudis, Panayotis Smyrnelis
TL;DR
The paper develops a unified variational framework to prove the existence of ground states in Hilbert spaces and to construct higher-dimensional ground states tied to heteroclinic orbits. By introducing an effective potential $\mathcal{W}$ on a Hilbert space and a renormalized energy $\mathcal{E}$, the authors anchor ground-state construction in radial variational problems and then lift one-dimensional heteroclinic structures to $d+k$ dimensions, producing layered, homoclinic-type solutions. Central tools include radial compactness arguments, Pohozaev-type identities, and energy renormalization to handle infinite-energy configurations. The results cover both $d\ge 3$ and $d=2$ cases, with detailed symmetry and nondegeneracy conditions ensuring the existence and minimality of the mapped solutions, offering a robust method for layered phase-transition elliptic systems and related PDEs.
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 $Δu(x)=\nabla W(u(x))$ (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 $u:\mathbb{R}^{d+1}\to\mathbb{R}^m$ ($d\geq 2$), 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.