Rafael Ortega and Gianmaria Verzini

We prove that, for every bounded and measurable forcing $p(t)$, the differential equation $\ddot{u}+u^{1/3} =p(t)$ has bounded solutions with arbitrarily large amplitude. In general it is not possible to say that all solutions are bounded, as shown by an example due to Littlewood.

The proof is based on a variational method which can be seen as a dual version of Nehari's method for boundary value problems on compact intervals.

2000 Mathematical Subject Classification: 34C11, 34B15, 49J35.

Keywords: bounded solution, minimax, Nehari's method.

E-mail:
rortega@ugr.es
gianmaria.verzini@polimi.it