Subharmonic solutions for second-order Hamiltonian systems
The existence of subharmonic solutions to the following second-order Hamiltonian system \[ u_{tt} + G'(u)= f(t), \tag{1} \] with \(G\in C^2(\mathbb{R}^N;\mathbb{R})\) and \( f: \mathbb{R} \to \mathbb{R}^N\) is a continuous function with a minimal period \(T>0\) and \(G'\) is the gradient of \(G\), is discussed in the article. A sequence \(\{u_{k_i}\}\) is called a subharmonic sequence for (1) if \(u_{k_i}\) is a periodic solution to (1) with the minimal period \(k_i T\) and \(k_i \to \infty\) as \( i \to \infty\). There are many investigations in which it was shown the existence of a subharmonic sequence for (1) under different assumptions on \(G\): 1) \(G\) is a convex functional; 2) \(G\) is a nonconvex functional (\(k_i\) - are prime numbers, \(G\) is bounded, \(G'\) has a linear growth); 3) \(G\) has a subquadratic growth and all critical points are nondegenerate. The existence of a subharmonic sequence for (1) in the case when each \(k_i\) is not necessarily a prime number and in the case when \(G\) has a degenerate critical point is proved in the article.
scientific article; zbMATH DE number 1492543