On a class of superlinear sturm Liouville problems with arbitrarily many solutions

We derive multiplicity results for autonomous superlinear ODE’s of the form $\begin{gathered} - u''(x) = g(u(x)) + t,\qquad x \in (0,\pi ),\quad t \in R, \hfill \\ u(0) = u(\pi ) = 0, \hfill \\ \end{gathered} $ with $g'( - \infty ) < + \infty $ and $g'( + \infty ) = + \infty $.We show that for any given $n \in N$ there exist at least n solutions of the problem if t is sufficiently negative. The proof is carried out by using variational methods jointly with a rearrangement argument.