Topological Methods in Nonlinear Analysis

Concentration of ground state solutions for fractional Hamiltonian systems

César Torres and Ziheng Zhang

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Abstract

We are concerned with the existence of ground states solutions to the following fractional Hamiltonian systems: \begin{equation} \begin{cases} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\ u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{cases} \tag*{$(\mbox{FHS})_\lambda$} \end{equation} 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 for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies the Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that (FHS)$_\lambda$ has a ground sate solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to $u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^n)$, where $u\in E_{0}^{\alpha}$ is a ground state solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Recent results are generalized and significantly improved.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 2 (2017), 623-642.

Dates
First available in Project Euclid: 11 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1507687547

Digital Object Identifier
doi:10.12775/TMNA.2017.033

Mathematical Reviews number (MathSciNet)
MR3747031

Zentralblatt MATH identifier
1383.34013

Citation

Torres, César; Zhang, Ziheng. Concentration of ground state solutions for fractional Hamiltonian systems. Topol. Methods Nonlinear Anal. 50 (2017), no. 2, 623--642. doi:10.12775/TMNA.2017.033. https://projecteuclid.org/euclid.tmna/1507687547


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