Variational approach to solutions for a class of fractional boundary value problem

Abstract

In this paper we investigate the existence of infinitely many solutions for the following fractional boundary value problem \begin{equation*} \begin{cases} _tD^{\alpha}_T(_0D^{\alpha}_t u(t))=\nabla W(t,u(t)),\qquad t\in [0,T],\\ u(0)=u(T)=0, \end{cases} \tag{FBVP} \end{equation*} where $\alpha\in (1/2,1)$, $u\in \mathbb{R}^n$, $W\in C^1([0,T]\times\mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. The novelty of this paper is that, assuming $W(t,u)$ is of subquadratic growth as $|u|\rightarrow+\infty$, we show that (FBVP) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved.

Keywords

• fractional hamiltonian systems
• critical point
• variational methods
• genus