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On the relativistic pendulum-type equation. (English) Zbl 07217165
In this paper, the authors consider the equation \[\Bigl(\frac{u'}{\sqrt{1-{u'}^2}}\Bigr)'+F'(u)=0\tag{1.1}\] and firstly look for solutions of the relativistic Dirichlet boundary value problem. They assume:
(\(F_1\)) \(F\in C^1(\mathbb{R})\) with \(F(0)=F'(0)=0\),
(\(F_2\)) \(F\) is \(C^2\) at \(u=0\) and \(F''(0)=\alpha^2>0\)
and prove the following theorem.
Theorem 3.1. Let (\(F_1-F_2\)) hold. If \(T>\frac{\pi}{\alpha}\), then (1.1) with \(u(0)=u(T)=0\) has at least two nontrivial solutions: \(u_1(t)>0\) on \((0,T)\) and \(u_2(t)<0\) on \((0,T)\).
This result is used to find non-trivial periodic solutions to (1.1).
Theorem 4.1. (i) Let (\(F_1-F_2\)) hold and let \(F\) be even. If \(T>\pi /\alpha\) equation (1.1) has a pair of periodic solutions \(\pm z_1\) with minimal period \(2T\).
(ii) If \(F\in C^2(\mathbb{R})\) and \(F''(u)\le\alpha^2,\forall u\in\mathbb{R}\) and \(T<\pi /\alpha\) the only \(2T\) periodic solution to (1.1) is \(u\equiv 0\).
In Section 5, the authors consider the equation \[\Bigl(\frac{u'}{\sqrt{1-{u'}^2}}\Bigr)'+F'(u)-h^2G'(u)=0,\] where \(h\) is the constant angular momentum and prove the existence of periodic solutions provided \(T\) is larger than a suitable value.
Finally, in Section 6, the authors prove the existence of multiple periodic solutions to the periodically forced spherical pendulum equation \[\Bigl(\frac{u'}{\sqrt{1-{u'}^2}}\Bigr)'+F'(u)-h^2G'(u)=f(t),\] where \(F\) and \(G\) are periodic.
The proofs are based on the critical point theory carried out by A. Szulkin [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 77–109 (1986; Zbl 0612.58011)] to handle functionals \(I\) which are not smooth.
34B15 Nonlinear boundary value problems for ordinary differential equations
49J40 Variational inequalities
35Q75 PDEs in connection with relativity and gravitational theory