| Volume 9, Number 6, 2019, Pages 2436-2453 DOI：10.11948/20190157 |
| Fractional Hamiltonian systems with positive semi-definite matrix |
| Cesar Enrique Torres Ledesma,Ziheng Zhang,Amado Mendez |
| Keywords:Fractional Hamiltonian systems, fractional Sobolev space, critical point theory, concentration phenomena. |
| Abstract: |
| We study the existence of solutions for the following fractional Hamiltonian systems
$$
\left\{
\begin{array}{ll}
- _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm]
u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
\end{array}
\right.
~~~~~~~~~~~~~~~~~(FHS)_\lambda
$$
where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that
$L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$,
$W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on
$\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution
of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$. |
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