Positive solutions of elliptic problems with locally oscillating nonlinearities

We prove the existence of two unbounded sequences of strictly positive solutions, obtained respectively as minimum and saddle points of the associated action functional, of the elliptic problem $$ \left\{ \begin{array}{cl} -\Delta u = f(x,u) & \mbox{ in }\, \Omega, \\ u = 0 & \mbox{ on }\, \partial \Omega, \end{array} \right. $$ assuming that an oscillatory behaviour at $+\infty$ of $s^{-2} \int_0^s f(x, \xi) d \xi$ occurs locally in $\Omega$.