具有Liouville-Weyl分数阶导数的分数阶Hamilton系统的多解性
Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivative
摘要：
本文研究了下列分数阶Hamilton系统：\begin{eqnarray}\label{eq00} _{t}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) + L(t)u(t) = & \nabla W(t,u(t))\\ u\in H^α(\mathbb{R}, \mathbb{R}^{N}).\nonumber \end{eqnarray}其中$α\in (1/2, 1)$，$t\in \mathbb{R}$，$u\in \mathbb{R}^{n}$，$L\in C(\mathbb{R}, \mathbb{R}^{n^2})$是对所有$t\in \mathbb{R}$，$W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$的对称正定矩阵，$\nabla W$是$W$在$u$的梯度.本文的新颖之处在于，假设存在$l\in C(\mathbb{R}, \mathbb{R})$使得对所有$t\in \mathbb{R}$，$u\in \mathbb{R}^{n}$和$l$上的下列条件满足$(L(t)u,u)\geq l(t)|u|^{2}$：$\inf_{t\in \mathbb{R}}l(t) >0$，存在$r_{0}>0$使得对任何$M>0$ $$ m(\{t\in (y-r_{0}, y+r_{0})/\;\;l(t)\leq M\}) \to 0\;\;\mbox{as}\;\;|y|\to \infty. $$都满足且$W$作为$|u| \to +\infty$次二次增长，我们利用临界理论中的亏格性质证明了（\ref{eq00}）具有无穷多个解. [Z. Zhang和R.袁，无强制条件下次二次分式哈密尔顿系统的解，数学方法应用科学，DOI：10.1002/mma.3031]显著改善。
英文摘要：
In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} _{t}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) + L(t)u(t) = & \nabla W(t,u(t))\\ u\in H^α(\mathbb{R}, \mathbb{R}^{N}).\nonumber \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$, and $\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming there exists $l\in C(\mathbb{R}, \mathbb{R})$ such that $(L(t)u,u)\geq l(t)|u|^{2}$ for all $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$ and the following conditions on $l$: $\inf_{t\in \mathbb{R}}l(t) >0$ and there exists $r_{0}>0$ such that, for any $M>0$ $$ m(\{t\in (y-r_{0}, y+r_{0})/\;\;l(t)\leq M\}) \to 0\;\;\mbox{as}\;\;|y|\to \infty. $$ are satisfied and $W$ is of subquadratic growth as $|u| \to +\infty$, we show that (\ref{eq00}) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in [Z. Zhang and R. Yuan, Solutions for subquadratic fractional Hamiltonian systems without coercive conditions, Math. Methods Appl. Sci., DOI: 10.1002/mma.3031] are significantly improved.
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