Existence and multiplicity results for periodic solutions of superquadratic Hamiltonian systems where the potential changes sign (Q1346277)

The paper deals with the existence of periodic solutions of the system (1) \(\ddot x + b(t) V'(x(t)) = 0\) where \(b(t)\) is a continuous \(T\)- periodic \((T > 0)\) real function and \(V\) has a superquadratic behavior. In case that \(b(t)\) changes its sign, some results have been obtained in other papers but under the assumption that \(V\) is homogeneous. Here the case \(\int^ T_ 0 b(t)dt \neq 0\) is considered and the homogeneity assumption on \(V\) is weakened, in the sense that one requires that \(V\) differs from a homogeneous superquadratic function by a quadratic term, outside a sufficiently large sphere. Several theorems which discuss the existence of periodic solutions of equation (1) are proved.