## 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 $$q\in W^{1,2}(R^ N,R)$$.

### MSC:

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

### Keywords:

periodic solution; Hamiltonian system; homoclinic solution