RETRACTED ARTICLE: Minimal Period Symmetric Solutions for Some Hamiltonian Systems Via the Nehari Manifold Method
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD∗ Chouha¨ıd SOUISSI Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000 -Monastir, Tunisia E-mail : [email protected] Abstract For a given T > 0, we prove, under the global ARS-condition and using the Nehari manifold method, the existence of a T -periodic solution having the W -symmetry introduced in [21], for the hamiltonian system z¨ + V ′ (z) = 0,
z ∈ RN ,
N ∈ N∗ .
Moreover, such a solution is shown to have T as a minimal period without relaying to any index theory. A multiplicity result is also proved under the same condition. Key words
Hamiltonian; variational; minimal period; Nehari Manifold
2010 MR Subject Classification
1
35J15; 35J20; 35J50
Introduction We consider the following second order hamiltonian system z¨ + V ′ (z) = 0,
(1.1)
where z : R → RN is a vector function, N is a positive integer, and V ′ is the gradient vector of the potential function V with respect to z. In [16], Rabinowitz proved the existence of non-constant periodic solutions with any prescribed period T > 0, for system (1.1). He supposed that V is superquadratic at the origin and satisfies the well known Ambrosetti-Rabinowitz’s condition ∃µ > 2,
r > 0,
s.t.
∀|z| > r,
0 < µV (z) ≤ zV ′ (z).
(AR)
He also conjectured that (1.1) possesses a non-constant solution with any prescribed minimal period. Later (1985), Ekeland and Hoffer [8] made a significant progress by confirming this conjecture for strictly convex hamiltonian systems. In the last years, many researchers has been based on (AR) in their proofs [1–6, 11, 13]. Few of them did not use any convexity hypotheses in their works. One can refer to Girardi, Matzeu [9, 10], Long [12], and Souissi [20, 21]. ∗ Received
February 6, 2018; revised April 21, 2018.
No.3
C. Souissi: MINIMAL PERIOD SYMMETRIC SOLUTIONS
615
In the last two references, the following new growth assumption was introduced, ∃n ∈ N∗ , ∃αn > 3 − n, such that V ∈ C n (RN , R) and αn V (n−1) (z)z n−1 ≤ V (n) (z)z n ,
∀z ∈ RN \ B,
(ARSn )
where V (n) is the nth derivative of V , for a given integer n ≥ 1. In [20], the author considered the global (ARSn )-condition B = {0} . He proved, thanks to the saddle point theorem, the existence of at least one even periodic solution for problem (1.1). Then, in [21], he used the local condition B = Br (0) and supposed the potential to be superquadratic at the origin. Using the Mountain-pass theorem, he proved the existence of at least one periodic solution of (1.1) which is not only even, but also satisfies the so called W -symmetry, introduced in that article and with which we will deal in the third section of this article. In both works, an iteration inequality on the symmetric Morse indices was used to study the minimal period of the solutions. This inequality is due to Long [12]. In [20], the solution has been shown to possess
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