On fractional Hamiltonian systems with indefinite sign sub-quadratic potentials. (Q2662473)

From the abstract: ``In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems \[\left\{\begin{array}{l}_tD_\infty^\alpha(_{-\infty}D_t^\alpha u(t))+L(t)u(t)=\nabla F(t,u(t)),\\ u\in H^\alpha(\mathbb{R},\mathbb{R}^n),\end{array}\right.\] where \(\alpha\in(1/2,1),u\in\mathbb{R}^N,L\in C(\mathbb{R},\mathbb{R}^{N\times N})\) and \(F\in C^1(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})\). The novelty of this paper is that, under the relaxed assumptions on \(F(t,x)\) and \(L(t)\), we obtain infinitely many solutions via genus properties in critical point theory.'' The results are motivated by the paper [\textit{C. E. Torres Ledesma}, Proc. Indian Acad. Sci., Math. Sci. 128, No. 4, Paper No. 50, 16 p. (2018; Zbl 1401.35324)] where the existence result was obtained under slightly different assumptions and with the use of the Mountain Pass theorem, and the paper [\textit{Z. Zhang} and \textit{R. Yuan}, Math. Methods Appl. Sci. 37, No. 18, 2934--2945 (2014; Zbl 1307.34019)] where the genus method was applied for the above system, under stronger assumptions on \(F\) and \(L\), to obtain infinitely many solutions. In the proof of the main theorem (Thm 2.2) the authors use the Rabinowitz criterium for the existence of critical points of functionals satisfying the PS-condition.