Concentration of ground state solution for a fractional Hamiltonian Systems

TL;DR

This paper proves the existence and concentration behavior of ground state solutions for a class of fractional Hamiltonian systems with variable coefficients, extending and improving recent results in the field.

Contribution

It establishes the existence of ground states for fractional Hamiltonian systems with variable coefficients and describes their concentration behavior as a parameter tends to infinity.

Findings

Ground state solutions exist for the fractional Hamiltonian system.

Solutions vanish outside a finite interval as the parameter increases.

Solutions converge to ground states of a related boundary value problem.

Abstract

In this paper we are concerned with the existence of ground states solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} -_tD^\alpha_\infty(_{-\infty}D^\alpha_t u(t)) - \lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u \in H^\alpha (\mathbb{R},\mathbb{R}^n), \end{array} \right.\qquad(\hbox{FHS})_\lambda$$ 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 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 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.