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In a series of papers the question of uniqueness of radial ground states of the equation ∆u + f(u) = 0 and of various related equations has been studied. It is remarkable that throughout this work (except in very special circumstances) nowhere is a spatially dependent term taken into consideration. Here we shall make a first attempt to study the uniqueness of ground states for such spatially dependent equations and to establish qualitative properties of solutions for this purpose. Mathematics Subject Classification (2000). Primary 35J70, Secondary 35J60

The existence of infinitely many subharmonic solutions is proved for the periodically forced nonlinear scalar equation u" + g(u) e(t), where g is a continuous function that is defined on a open proper interval (A, B) C ]. The nonlinear restoring field g is supposed to have some singular behaviour at the boundary of its domain. The following two main possibilities are analyzed:(a) The domain is unbounded and g is sublinear at infinity. In this case, via critical point theory, it is possible to prove the existence of a sequence of subharmonics whose amplitudes and minimal periods tend to infinity.(b) The domain is bounded and the periodic forcing term e(t) has minimal period T 0. In this case, using the generalized Poincar-Birkhoff fixed point theorem, it is possible to show that for any m E 1, there are infinitely many periodic solutions having mT as minimal period.Applications are given to the dynamics of a charged particle moving on a line over which one has placed some electric charges of the same sign.

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