Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign. (English) Zbl 0771.34031

This paper considers the Hamiltonian system \((*)\) \(\ddot q=B(t)q+b(t)W'(q)=0\) \((q\in R^ N)\) with the following assumptions: \((\text{V}_ 1)\) \(B\) is a continuous, \(T\)-periodic, positive definite and symmetric matrix valued function; \((\text{V}_ 2)\) \(b\) is a continuous \(T\)-periodic real function, and there exist \(t_ 1\), \(t_ 2\) such that \(b(t_ 1)>0\) and \(b(t_ 2)<0\); \((\text{V}_ 3)\) \(W\in C'(R^ N,R)\), \(W(x)>0\) \(\forall x\neq 0\) and there exists \(\mu>2\) such that \(W(\lambda x)=\lambda^ \mu W(x)\) \(\forall\lambda\geq 0\) and \(x\in R^ N\). The main results consist of the following theorems: 1) Assume \((\text{V}_ 1)-(\text{V}_ 2)-(\text{V}_ 3)\). Then \((*)\) has at least one nonconstant \(T\)-periodic solution; 2) Assume \((\text{V}_ 1)- (\text{V}_ 2)-(\text{V}_ 3)\), and moreover, \(W\in C^ 2(R^ N,R)\) is even. Then \((*)\) has infinitely many (pairs of) nonconstant \(T\)- periodic solutions; 3) Assume \((\text{V}_ 1)-(\text{V}_ 2)- (\text{V}_ 3)\). Then \((*)\) possesses a homoclinic solution \(q(t)\), emanating from zero such that \(q\in W^{1,2}(R^ N,R)\).


34C25 Periodic solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations