Multiple solutions for a class of superquadratic fractional Hamiltonian systems

Authors : Mohsen Timoumi

Pages : 186-195

Doi:10.32323/ujma.388067

View : 11 | Download : 7

Publication Date : 2018-09-30

Article Type : Research Paper

Abstract :In this paper, we are concerned with the existence of solutions for a class of fractional Hamiltonian systems \[\left\{ \begin{array}{l} _{t}D_{\infty}^{\alpha}insert ignore into journalissuearticles values(_{-\infty}D_{t}^{\alpha}u);insert ignore into journalissuearticles values(t);+Linsert ignore into journalissuearticles values(t);uinsert ignore into journalissuearticles values(t);=\nabla Winsert ignore into journalissuearticles values(t,uinsert ignore into journalissuearticles values(t););,\ t\in\mathbb{R}\\ u\in H^{\alpha}insert ignore into journalissuearticles values(\mathbb{R},\ \mathbb{R}^{N});, \end{array}\right. \] where $_{t}D_{\infty}^{\alpha}$ and $_{-\infty}D^{\alpha}_{t}$ are the Liouville-Weyl fractional derivatives of order $\frac{1}{2}<\alpha<1$, $L\in Cinsert ignore into journalissuearticles values(\mathbb{R},\mathbb{R}^{N^{2}});$ is a symmetric matrix-valued function and $Winsert ignore into journalissuearticles values(t,x);\in C^{1}insert ignore into journalissuearticles values(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R});$. Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many solutions for insert ignore into journalissuearticles values(1); when $L$ is not required to be either uniformly positive definite or coercive and $Winsert ignore into journalissuearticles values(t,x);$ satisfies some weaker superquadratic conditions at infinity in the second variable but does not satisfy the well-known Ambrosetti-Rabinowitz superquadratic growth condition.
Keywords : Fractional Hamiltonian systems, Variational methods, Symmetric Mountain Pass Theorem