A topological approach to superlinear indefinite boundary value problems (Q1590108)

Some boundary value problems are considered for the differential equation \(\ddot x + q(t)g(x) = 0\), where \(g(x)\) has superlinear growth at infinity and \(q(t)\) changes sign. It is shown, that there exist solutions with a prescribed number of zeroes in each interval of positivity of \(q(t)\). Moreover, for each interval of negativity, one can arbitrarily specify in advance that the solution has exactly one zero being also strictly monotone or that it has no zeroes and exactly one zero of the first derivative is in this interval. Various generalizations are discussed.