Homoclinics: Poincaré-Melnikov type results via a variational approach. (English) Zbl 1004.37043

From the introduction: This paper deals with the existence of homoclinics, namely doubly-asymptotic solutions, for a broad class of perturbed differential equations, variational in nature.
The existence of homoclinics has been faced both from the local and from the global point of view. The existence for perturbed time periodic systems with one degree of freedom was first proved by Poincaré. The results by Poincaré have been the starting point for a great deal of work. In particular, Melnikov has proved by analytical methods the existence of homoclinics for non conservative perturbations, leading to chaos. A common feature of these results is the use of an integral function, the Poincaré function or – roughly – its derivative, the Melnikov function. The non degenerate zeros of the latter give rise to homoclinics.
On the other side, more recently Critical Point Theory has been used to prove the existence of homoclinics for a class of Hamiltonian systems like \[ \ddot u-u+\nabla U(t,u)=0, \] when the potential \(U\simeq|u |^{p+1}\) with \(p>1\) and depends periodically (or almost periodically) on \(t\). Although these approaches are apparently considered different in nature, we show that they are connected, in the sense that an appropriate use of Critical Point Theory permits also to find the classical perturbation results. More precisely, we discuss an approach, variational in nature, which furnishes a general frame to deal with several different kinds of perturbed differential equations. We not only find Poincaré-Melnikov like results (both for systems with several degrees of freedom and for autonomous systems) without any non degeneracy assumption, but also handle partial differential equations.
Moreover, the recent variational works cannot provide results in the same generality: we localize the solutions and find multiplicity results; in addition, we can also handle potentials with a more general dependence on \(t\), not only periodic or almost periodic.
In order to have an idea of our setting, let us consider the second-order Hamiltonian system with \(N\) degrees of freedom \[ \ddot u-u+\nabla V(u)=\varepsilon \nabla_uW(t,u),\tag{1} \] where \(V(0)=0\), \(\nabla V(0)=0\), \(D^2V(0)=0\), and roughly \(W(t,0)=0\), \(\nabla_u W(t,0)=0\). Homoclinics of (1) correspond to stationary points \(u\in W^{1,2} (\mathbb{R})\) of the Lagrangian functional \[ \int_\mathbb{R} [L_0+\varepsilon W],\;L_0 (x,p)=\frac 12\bigl[ |p|^2+|x|^2 \bigr]+V(x), \] whose Euler equation is (1). Suppose that the unperturbed equation \[ \ddot u-u+\nabla V(u) =0 \] has a non trivial homoclinic \(u_0(t)\). Connected with \(u_0\), the unperturbed functional \(\int L_0\) possesses a manifold of critical points \(Z= \{u_0(t+\theta): \theta\in\mathbb{R}\}\) and we are led to search homoclinics near one of these translates \(u_0(\cdot+ \theta)\) by looking for critical points of \(\int L_0+\varepsilon\int W\) nearby \(Z\). It turns out that these critical points exist provided that \[ \Gamma(\theta)= \int_\mathbb{R} W\bigl(t,u_0 (t+\theta)\bigr) dt, \] has a (possibly degenerate) critical point. Such a \(\Gamma\) is the Poincaré function and its derivative is the Melnikov function.
Section 2 deals with the existence of stationary points for a class of functionals like \[ f_\varepsilon(u)= f_0(u)+\varepsilon G(u). \] This is applied in Section 3 to (1) as well as to perturbed radial systems like \[ \ddot u-u+|u|^{p-1} u=\varepsilon \nabla_u W(t,u),\;p>1.\tag{2} \] In this latter case \(Z=\{\xi r(t+ \theta)\} \simeq S^{N-1}\times \mathbb{R}\) where \(\xi\in S^{N-1}\) and \(r>0\) satisfies \[ \ddot r-r+r^p=0,\;\lim_{t\to\pm \infty}r(t)=0. \] One shows that the condition \(T_zZ=\text{Ker}[f_0''(z)]\) is still satisfied and hence the abstract approach yields the existence of homoclinics in connection with the critical points of \[ \Gamma(\xi,\theta)= \int_\mathbb{R} W\bigl(t,\xi r(t+\theta) \bigr)dt,\;(\xi,\theta)\in S^{N-1} \times\mathbb{R}. \] As for the perturbation \(W\), we can also consider the case that \(W(t,u)=g(t)\circ u\), when (1) becomes a forced system. In particular, the classical case of systems with a periodic or quasi periodic forcing term is handled in Section 4.
The functional approach also applies when \(W\) is independent of time and (2) becomes an autonomous system (see Section 5). In such a case we can show the existence of two distinct homoclinics, a multiplicity result which improves the one of [K. Tanaka, NoDEA 1, 149-162 (1994; Zbl 0819.34032)]. The generality of our abstract setting allows us to handle partial differential equations, too. Applications to the existence of semiclassical states of a class of Schrödinger equations with potential have been discussed in [A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of Nonlinear Schrödinger equations, Arch. Rations Mech. Anal. (to appear)]. Here, see Section 6, we prove the existence of two solutions of forced Schrödinger equations like \[ \begin{cases} -\Delta u+u=|u|^{p-1}u-\varepsilon g(x),\;x\in\mathbb{R}^N,\\ u(x)\to 0\quad\text{ as }|x|\to \infty.\end{cases} \] provided \(1<p<(N+2)/(N-2)\) and \(g\in L^2\), improving a recent result of [L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations, (preprint.)]
The authors’ technique has been extended to obtain results of multibump type [M. Berti and P. Bolle, Ann. Mat. Pura Appl. (4) 176, 323-378 (1999; Zbl 0957.37019)]; and to a nonvariational setting [M. Henrard, Discrete Contin. Dyn. Syst. 5, No. 4, 765-782 (1999; Zbl 0980.34043)].


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C29 Homoclinic and heteroclinic orbits for dynamical systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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