Periodic Solutions for N -Body-Type Problems
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Periodic Solutions for N -Body-Type Problems∗ Fengying LI1
Shiqing ZHANG2
Abstract The authors consider non-autonomous N -body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity. Using LjusternikSchnirelmann theory, the authors prove the existence of unbounded sequences of critical values for the Lagrangian action corresponding to non-collision periodic solutions. Keywords Periodic solutions, N -body type problems, Variational methods 2000 MR Subject Classification 34C15, 34C25, 58F, 70F07
1 Introduction and Main Results The 1975 paper by Gordon [1] exhibits the first prominent use of variational methods in the study of periodic solutions of the following Newtonian equations with singular potential V (t, x) ∈ C 1 ([0, T ] × (Rn \S), R), ( x¨ + V ′ (t, x) = 0, x ∈ Rn \S, (1.1) x(t + T ) = x(t), where the potential V (t, x) satisfies V (t + T, x) = V (t, x) and Gordon’s strong-force (SF for short) condition which stipulates that there exists a neighbourhood N of the compact set S and a function U ∈ C 2 (N \S, R) such that (i) U (x) → −∞ as x → S; (ii) −V (t, x) ≥ |∇U (x)|2 , ∀x ∈ N \S. −a |x|α (a > 0, α ≥ 2), and a 2 |x|2 = |∇U (x)| when |x| ≤
Remark 1.1 For a simple example, let V (t, x) =
√ a ln |x|. Then ∇U (x) =
√
ax |x|2
and −V (t, x) =
a |x|α
≥
take U (x) = 1.
The function U (x) is introduced to control the potential V (t, x) and force the Lagrange functional of the system (1.1) to satisfy the Palais-Smale condition. This is a significant step in utilizing the calculus of variations to obtain the following result. Theorem 1.1 (Gordon) Under the above conditions and the following condition (G1 ) : V (t, x) < 0, x = 6 0, Manuscript received March 3, 2018. of Economics and Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China. E-mail: [email protected] 2 Department of Mathematics, Sichuan University, Chengdu 610064, China. E-mail: [email protected] ∗ This work was supported by the National Natural Science Foundation of China (Nos. 11701463, 11671278). 1 School
734
F. Y. Li and S. Q. Zhang
there exist periodic solutions of (1.1) which tie (wind around) S and have arbitrary given topological (homotopy) type and given period. Ambrosetti-Coti Zelati [2–3] used Morse theory to generalize Gordon’s result and obtained the following theorem. Theorem 1.2 Assume V ∈ C 2 ([0, T ] × Rn , R) satisfies V (t + T, x) = V (t, x), Gordon’s strong force condition and the following condition: (A) : |V (t, x)|, |Vx (t, x)| → 0 uniformly for all t as kxk → ∞, and ∃R1 > 0 such that V (t, x) < 0, ∀kxk ≥ R1 , then (1.1) has infinitely many T -periodic solutions. Motivated by Gordon [1] and Ambrosetti-Coti Zelati [2–3], Jiang [4] applied Morse theory and proved the existence of infinitely many periodic solutions using a weaker condition than above condition (A), proving the following result. Theorem 1.3 Let Ω be an open subset in
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