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Periodic solutions for a class of second-order Hamiltonian systems. (English) Zbl 1096.34027

Consider the eigenvalue problem for the second-order Hamiltonian system \[ \ddot u=\nabla_u\,F(t,u)\quad\text{ a.e. for }\quad t\in [0,\,T],\quad u(T)-u(0)=\dot u(T)-\dot u(0)=0\,. \]
H. BrĂ©zis and L. Nirenberg [Commun. Pure Appl. Math. 44, 939–963 (1991; Zbl 0751.58006)] established that this problem admits three periodic solutions under the assumptions:
a) \(F(t,0)=0\), \(\nabla_u\,F(t,0)=0\);
b) \(\lim_{| u| \to +\infty} F(t,u)=+\infty\) uniformly in \(t\);
c) For some constant vector \(u_0\), \[ \int_0^T F(t,u_0)\,dt< \int_0^T F(t,0)\,dt; \]
d) There exist \(r>0\) and an integer \(k\geq 0\) such that
\[ -{1\over 2}\,(k+1)^2 w^2 | u| ^2\leq F(t,u)-F(t,0)\leq -{1\over 2}\,k^2 w^2 | u| ^2\,, \]
for all \(| u| \leq r\), a.e. for \(t\in[0,\,T]\), where \(w=2\pi/T\).
It was proved by C. Tang and X. Wu [J. Math. Anal. Appl. 259, 386–397 (2001; Zbl 0999.34039)] that this problem possesses at least one periodic solution, when instead of d) a weaker coercivity-type condition is assumed. The present paper proves that the considered problem still possesses three periodic solutions when neither condition d) nor the weaker coercivity type condition are assumed. Further relaxation of the hypotheses are also discussed. The results obtained are based on the critical point theorems, the mountain pass theorem and some variational type principles.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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