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Liouville-Weyl Fractional Hamiltonian Systems: Existence Result |
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PP: 207-215 |
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doi:10.18576/pfda/050303
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Author(s) |
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Cesar Torres,
Willy Zubiaga,
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Abstract |
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| In this work we investigate the following fractional Hamiltonian systems
%\begin{eqnarray}\label{eq00}
$_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = \nabla W(t,u(t))$,
%\end{eqnarray}
where $\alpha \in (1/2, 1)$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a positive definite symmetric matrix, $W(t,u) = a(t)V(t)$ with $a\in C(\mathbb{R},\mathbb{R}^{+})$ and $V\in C^{1}(\mathbb{R}^{n}, \mathbb{R})$. By using the Mountain pass theorem and assuming that there exist $M>0$ such that $(L(t)u,u)\geq M|u|^{2}$ for all $(t,u)\in \mathbb{R}\times \mathbb{R}^{n}$ and $V$ satisfies the global Ambrosetti-Rabinowitz condition and other suitable conditions, we prove that the above mentioned equation at least has one nontrivial weak solution. |
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