Abstract
Some existence and multiplicity of periodic solutions are obtained for nonautonomous second order Hamiltonian systems with sublinear nonlinearity by using the least action principle and minimax methods in critical point theory.Mathematics Subject Classification (2000): 34C25, 37J45, 58E50.
Highlights
Introduction and main resultsConsider the second order systems u(t) = ∇F(t, u(t)) a.e. t ∈ [0, T], u(0) − u(T) = u (0) − u (T) = 0, (1:1)where T > 0 and F : [0, T] × RN ® R satisfies the following assumption: (A) F (t, x) is measurable in t for every x Î RN and continuously differentiable in x for a.e. t Î [0, T], and there exist a Î C(R+, R+), b Î L1(0, T ; R+) such that|F(t, x)| ≤ a(|x|)b(t), |∇F(t, x)| ≤ a(|x|)b(t) for all x Î RN and a.e. t Î [0, T]
| u(t)|2dt (Wirtinger’s inequality), where ||u||∞ := 0m≤ta≤xT|u(t)|. It follows from assumption (A) that the corresponding function on HT1 given by φ(u) :=
Which implies that φ(u) = 1 |u(t)|2dt + [F(t, u(t)) − F(t, u)]dt+ F(t, u)dt− F(t, 0)dt 2
Summary
It follows from assumption (A) that the corresponding function on HT1 given by φ(u) :=. Suppose that there exists a positive function h which satisfies the conditions (i), (iii), (iv) of (S1), we have the following estimates:. For u ∈ HT1, it follows from (S1), Lemma 2.1 and Sobolev’s inequality that [F(t, u(t)) − F(t, u)]dt. In order to apply the saddle point theorem in [2,3], we only need to verify the following conditions:. ≤ εC10 |u(t)|2dt + C11⎝ |u (t)|2dt⎠ , which implies that φ(u) = 1 |u(t)|2dt + [F(t, u(t)) − F(t, 0)] dt 2. Ψ Î C1(E, R) satisfies the (PS) condition by the proof of Theorem 1.2.
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