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Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials |
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PP: 231-240 |
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doi:10.18576/pfda/100204
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Author(s) |
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Wafa Selmi,
Mohsen Timoumi,
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Abstract |
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| In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian system
\begin{equation}
\label{eq1}
\left\{
\begin{array}{l}
_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R}\u\in H^{\alpha}(\mathbb{R}),
\end{array}\right.
\end{equation}
where $_{-\infty}D_{t}^{\alpha}$ and $_{t}D^{\alpha}_{\infty}$ are left and right Liouville-Weyl fractional derivatives of order $\frac{1}{2}<\alpha<1$ on the whole axis respectively, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix valued function unnecessary coercive and $W(t,x)\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below and unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that (\ref{eq1}) possesses infinitely many solutions via a variant Symmetric Mountain Pass Theorem. |
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