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For the classical N-body problem in R d with d ≧ 2 , Maderna–Venturelli in their remarkable paper (Ann Math 192:499–550, 2020) proved the existence of hyperbolic motions with any positive energy constant, starting from any configuration and along any non-collision configuration. Their original proof relies on the long time behavior of solutions by Chazy 1922 and Marchal-Saari 1976, on the Hölder estimate for Mañé’s potential by Maderna 2012, and on the weak KAM theory. We give a new and completely different proof for the above existence of hyperbolic motions. The central idea is that, via some geometric observation, we build up uniform estimates for Euclidean length and angle of geodesics of Mañé’s potential starting from a given configuration and ending at the ray along a given non-collision configuration. Moreover, our geometric approach works for Hamiltonians 1 2 ‖ p ‖ 2 - F (x) , where F (x) ≧ 0 is lower semicontinuous and decreases very slowly to 0 faraway from collisions. We therefore obtain the existence of hyperbolic motions to such Hamiltonians with any positive energy constant, starting from any admissible configuration and along any non-collision configuration. Consequently, for several important potentials F ∈ C 2 (Ω) , we get similar existence of hyperbolic motions to the generalized N-body system x ¨ = ∇ x F (x) , which is an extension of Maderna–Venturelli [Ann Math 2020]. [ABSTRACT FROM AUTHOR]

In this paper, we study the following logarithmic Schrödinger equation: - Δ u + a (x) u = u log u 2 in V , where Δ is the graph Laplacian, G = (V , E) is a connected locally finite graph, the potential a : V → R is bounded from below and may change sign. We first establish two Sobolev compact embedding theorems in the case when different assumptions are imposed on a(x). This leads to two kinds of associated energy functionals, one of which is not well defined under the logarithmic nonlinearity, while the other is C 1 . The existence of ground state solutions are then obtained by using the Nehari manifold method and the mountain pass theorem respectively. [ABSTRACT FROM AUTHOR]

We are concerned with the following nonlinear Schrödinger equation: - Δ u + λ u = f (u) in R 2 , u ∈ H 1 (R 2) , ∫ R 2 u 2 d x = ρ , where ρ > 0 is given, λ ∈ R arises as a Lagrange multiplier and f satisfies an exponential critical growth. Without assuming the Ambrosetti–Rabinowitz condition, we show the existence of normalized ground state solutions for any ρ > 0 . The proof is based on a constrained minimization method and the Trudinger–Moser inequality in R 2 . [ABSTRACT FROM AUTHOR]

It is shown that each lobe of the figure-eight orbit is star-shaped, which indicates that the corresponding polar angle is monotone. In general, it is not clear under which assumptions a trajectory in a minimizer is star-shaped. In this paper, we study minimizers connecting two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with two equal masses. We show that if the Jacobi coordinates of the two fixed-ends are both orthogonal, then for any minimizer, both Jacobi vectors describe a star-shaped curve. If the Jacobi coordinates are orthogonal only on one of the fixed-ends and they are in adjacent closed quadrants, then their polar angles have at most one critical point. As an application, we show the existence and prove some geometric properties of two families of periodic orbits. [ABSTRACT FROM AUTHOR]

For each given n ≥ 2 , we study the variational existence of a new spatial periodic orbit in the 2n-body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2n-body problem. Furthermore, with the help of numerical calculation, the orbit is spatial in the equal-mass four-body problem. [ABSTRACT FROM AUTHOR]

This paper gives an analytic existence proof of the Schubart periodic orbit with arbitrary masses, a periodic orbit with singularities in the collinear three-body problem. A 'turning point' technique is introduced to exclude the possibility of extra collisions and the existence of this orbit follows by a continuity argument on differential equations generated by the regularized Hamiltonian. [ABSTRACT FROM AUTHOR]

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53. [ABSTRACT FROM AUTHOR]

This paper gives an analytic proof of the existence of Schubart-like orbit, a periodic orbit with singularities in the symmetric collinear four-body problem. In each period of the Schubart-like orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proved by using a 'turning point' technique and a continuity argument on differential equations of the regularized Hamiltonian. [ABSTRACT FROM AUTHOR]

A correction to the article "Existence and Stability of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem" that was published in the March 31, 2012 online issue is presented.