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Three periodic solutions for perturbed second order Hamiltonian systems. (English) Zbl 1185.34048
Consider the following problem
\[ \begin{cases} -\ddot{u}+A(t)u=\nabla F(t,u)+\lambda\nabla G(t,u) \qquad \text{a.e. in } [0,T],\\ u(T)-u(0)=\dot{u}(T)-\dot{u}(0)=0, \end{cases}\tag{1} \] where \(\lambda\in \mathbb{R}\), \(T\) is a positive real number. \(A : [0,T] \rightarrow \mathbb{R}^{N\times N} \) is a continuous map from the interval \([0,T]\) to the set of \(N\)-order symmetric matrices, \(F,G :[0,T]\times\mathbb{R}^N \rightarrow \mathbb{R}\) are measurable with respect to \(t\) for every \(x\in \mathbb{R}^N\), continuously differentiable in \(x\) for almost every \(t\in [0,T]\) and satisfy the following condition:
\[ \sup_{|x|\leq c}\max\{|F(.,x)|,|G(.,x)|,|\nabla F(.,x)|,|\nabla G(.,x)|\}\in L^1[0,T] \] for all \(c\geq 0\). Assume that
\(\lim_{|x| \to +\infty}\left(\frac{1}{2}\lambda_1(A)|x|^2-F(t,x)\right)=+\infty\) uniformly in [0,T], where
\[ \lambda_1(A)=\inf_{u\in H_T^1,\|u\|=1}\left(\int_0^T\left(|\dot{u}(t)|^2 +\left(A(t)u(t),u(t)\right)\right)dt\right); \]
there exists \(\delta>0\) such that \(\frac{1}{2}\lambda_1(A)|x|^2-F(t,x)>0\) for all \(x\in \mathbb{R}^N \backslash \{0\}\) with \(|x|<\delta\) and a.e. \(t\in \text{[0,T]}\);
There exists \(x_0\in \mathbb{R}^N\) such that \(\int_0^T(A(t)x_0,x_0)dt<\int_0^T F(t,x_0)dt\).
Then the authors prove that there exist \(\lambda^*>0\) and \(r>0\) such that, for every \(\lambda\in ]-\lambda^*, \lambda^*[\), problem (1) admits at least three distinct solutions which belong to \(B_r(0)\subseteq H_T^1\) by using variational methods.

34C25 Periodic solutions to ordinary differential equations
47J30 Variational methods involving nonlinear operators
Full Text: DOI
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