Multiplicity of solutions for nonperiodic perturbed fractional Hamiltonian systems

• Main Author: Abderrazek Benhassine
• Format: Article
• Language: English
• Published: Texas State University 2017-03-01
• Series: Electronic Journal of Differential Equations
• Subjects: Fractional Hamiltonian systems; critical point; variational methods
• Online Access: http://ejde.math.txstate.edu/Volumes/2017/93/abstr.html

In this article, we prove the existence and multiplicity of nontrivial solutions for the nonperiodic perturbed fractional Hamiltonian systems $$\displaylines{ -_{t}D^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}x(t)) -\lambda L(t)\cdot x(t)+\nabla W(t,x(t))=f(t),\cr x\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}), }$$ where $\alpha \in (1/2 , 1],\; \lambda> 0 $ is a parameter, $t\in \mathbb{R}, x\in \mathbb{R}^N$, ${}_{-\infty}D^{\alpha}_{t}$ and ${}_{t}D^{\alpha}_{\infty}$ are left and right Liouville-Weyl fractional derivatives of order $\alpha$ on the whole axis $\mathbb{R}$ respectively, the matrix $L(t)$ is not necessary positive definite for all $t\in \mathbb{R}$ nor coercive, $W \in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$ and $f\in L^{2}(\mathbb{R},\mathbb{R}^{N})\backslash\{0\}$ small enough. Replacing the Ambrosetti-Rabinowitz Condition by general superquadratic assumpt ions, we establish the existence and multiplicity results for the above system. Some examples are also given to illustrate our results.