Existence of connecting orbits in a potential well. (English) Zbl 0817.34029

The author considers the well studied class of equations \(\ddot q+ V'(q)= 0\), \(q\in \mathbb{R}^ N\) under the assumption that the potential \(V(q)\) has a local maximum at \(0\in \mathbb{R}^ N\) contained in the open set \(\Omega= \{x\in \mathbb{R}^ N: V(x)< 0\}\cup\{0\}\neq \mathbb{R}^ N\). The interest lies in the existence of non-trivial \(C^ 2\) solutions which are either homoclinic at the origin, touching the boundary of \(\Omega\), or heteroclinic, joining the origin to the boundary. When \(\Omega\) is bounded and \(V'(x)\neq 0\), \(\forall x\in \partial\Omega\) homoclinic solutions have been extensively studied [A. Ambrosetti and M. L. Bertotti, Pitman Res. Notes Math. Ser. 269, 21-37 (1992; Zbl 0804.34046); P. H. Rabinowitz and K. Tanaka, Math. Z. 206, 473-499 (1991; Zbl 0716.58013)]. The second condition automatically rules out the existence of heteroclinic orbits. In this paper it is shown that even when \(V'\equiv 0\) on \(\partial\Omega\) there is a large class of potentials with homoclinic solutions. Heteroclinic orbits are no longer disallowed and mild conditions are given for the existence of one of either of these types. The restriction that \(\Omega\) be bounded is also removed, following an approach of P. Caldiroli [Proc. Roy. Soc. Edinburgh (to appear)].
Finally more general results still are given: the existence of solutions connecting disjoint components of \(\mathbb{R}^ N\backslash \Omega\).


34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces