Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface (Q2883260)

This article is devoted to second-order Hamiltonian systems NEWLINE\[NEWLINE -\ddot{q}(t) = V'(q(t))\,, \tag{1}NEWLINE\]NEWLINE where \(\ddot{q}(t)\) is the second-order derivative of \(q\) with respect to \(t\)\,, \(V\!\in\! C^1(\mathbb R^N,\mathbb R)\,,\) and \(V'\) denotes the gradient of \(V\) with respect to \(x\)\,. The first integral of system (1) is \(H(p,q)=\frac{1}{2}|p|^2+V(q),\;p=\dot q\).NEWLINENEWLINEThe authors investigate whether (1) has a periodic solution on a fixed energy surface \(\{(p,q): H(p,q)=h\}.\)NEWLINENEWLINEThe main result is Theorem 1.3.NEWLINENEWLINESuppose \(V\in C^1(\mathbb R^n,\mathbb R)\) satisfies the conditionsNEWLINENEWLINE(V1) \(V\) achieves a global minimum \(V_0\) at \(x_0\); andNEWLINENEWLINE(V2) \(\mathcal{H} := \liminf_{|x|\to\infty} V(x) >V_0\).NEWLINENEWLINEThen, for almost all \(h\in (V_0, \mathcal{H})\), there exists a non-constant periodic solution of energy \(h\).NEWLINENEWLINETo prove the existence of a solution on all energy surfaces the authors suppose thatNEWLINENEWLINE(A1) there exist positive constants \(\mu_1\) and \(\mu_2\) such that NEWLINE\[NEWLINE(V'(x),x) \geq \mu_1 V(x) - \mu_2\qquad \forall x\in \mathbb R^N.NEWLINE\]NEWLINE They obtain the following theorem.NEWLINENEWLINETheorem 1.4. Suppose \(V\in C^1(\mathbb R^N,\mathbb R)\) satisfies (V1), (V2) and (A1). Then, for any \(h\in (\mu_2/\mu_1, \mathcal{H})\), there exists a non-constant periodic solution of energy \(h\).NEWLINENEWLINEThe authors apply the functional NEWLINE\[NEWLINE f_h(u) = \frac{1}{2} \int\limits_0^1 |\dot{u}|^2 \, dt \int\limits_0^1 (h-V(u)) \, dt NEWLINE\]NEWLINE and use the monotonicity method to treat this functional and to obtain the solution to (1).