Electron. J. Diff. Equ., Vol. 2016 (2016), No. 40, pp. 1-13.

Existence of solutions to fractional Hamiltonian systems with combined nonlinearities

Ziheng Zhang, Rong Yuan

Abstract:
This article concerns the existence of solutions for the fractional Hamiltonian system
$$\displaylines{ - _tD^{\alpha}_{\infty}\big(_{-\infty}D^{\alpha}_{t}u(t)\big) -L(t)u(t)+\nabla W(t,u(t))=0,\cr u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), }$$
where $\alpha\in (1/2,1)$, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and positive definite matrix. The novelty of this article is that if $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ and the nonlinearity $W(t,u)$ involves a combination of superquadratic and subquadratic terms, the Hamiltonian system possesses at least two nontrivial solutions.

Submitted November 11, 2015. Published January 27, 2016.
Math Subject Classifications: 34C37, 35A15, 35B38.
Key Words: Fractional Hamiltonian systems; critical point; variational methods; mountain pass theorem.

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Ziheng Zhang
Department of Mathematics
Tianjin Polytechnic University
Tianjin 300387, China
email: zhzh@mail.bnu.edu.cn
Rong Yuan
Department of Mathematical Sciences
Beijing Normal University
Beijing 100875, China
email: ryuan@bnu.edu.cn

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