Sufficient conditions on the non-linearity f are given which ensure that non-trivial solutions of second order differential equations of the form Lu = f(t, u) have a finite number of transverse zeros in a given finite time interval. We also obtain a priori lower bounds on the separation of zeros of solutions. In particular our results apply to non-Lipschitz non-linearities. Applications to non-linear porous medium equations are considered, yielding information on the existence and strict positivity of equilibrium solutions in some important classes of equations.
Laister, R., Laister, R., & Beardmore, R. E. (2003). Transversality and separation of zeros in second order differential equations. Proceedings of the American Mathematical Society, 131(1), 209-218. https://doi.org/10.1090/S0002-9939-02-06546-2