Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign

Abstract

In this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form tD∞α(-∞Dtαu(t))+L(t)u(t)=∇W(t,u(t))${\,}_tD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}u(t))+L(t)u(t)=\nabla W(t,u(t))$, where α∈(12,1)${\alpha \in (\frac{1}{2},1)}$, t∈ℝ${t\in \mathbb {R}}$, u∈ℝn${u\in \mathbb {R}^n}$, L∈C(ℝ,ℝn2)${L\in C(\mathbb {R},\mathbb {R}^{n^2})}$ is a symmetric and positive definite matrix for all t∈ℝ${t\in \mathbb {R}}$, W∈C1(ℝ×ℝn,ℝ)${W\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R})}$ and ∇W(t,u)${\nabla W(t,u)}$ is the gradient of W(t,u)${W(t,u)}$ at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants 0<τ1<τ2<∞${0<\tau _1<\tau _2< \infty }$ such that τ1|u|2≤(L(t)u,u)≤τ2|u|2${\tau _1 |u|^2\le (L(t)u,u)\le \tau _2 |u|^2}$ for all (t,u)∈ℝ×ℝn${(t,u)\in \mathbb {R}\times \mathbb {R}^n}$ and W(t,u)${W(t,u)}$ is of the form (a(t)/(p+1))|u|p+1${({a(t)}/({p+1}))|u|^{p+1}}$ such that a∈L∞(ℝ,ℝ)${a\in L^{\infty }(\mathbb {R},\mathbb {R})}$ can change its sign and 0<p<1${0<p<1}$ is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.

Keywords

• fractional hamiltonian systems
• critical point
• variational methods
• symmetric mountain pass theorem
• 34c37
• 35a15
• 35b38