Perturbation from Dirichlet problem involving oscillating nonlinearities

In this paper we prove that if the potential $F(x,t)=\int_0^tf(x,s)ds$ has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), then under very mild conditions on the perturbation term $g$ for every $k\in N$ there exists $b_k > 0$ such that \begin{displaymath} \left\{ \begin{array}{ll} -\Delta u = f(x,u)+\lambda g(x,u) & \mbox{in\ } \Omega \\ u=0 & \mbox{on\ } \partial \Omega \end{array}\right. \end{displaymath} has at least $k$ distinct weak solutions in $W_0^{1,2}(\Omega)$, for every $\lambda\in \mathbb{R}$ with $|\lambda|\leq b_k$. Moreover, information about the location of such solutions is also given. In fact, there exists a positive real number $\sigma>0$, which does not depend on $\lambda$, such that the $W_0^{1,2}(\Omega)$ norm of each of those $k$ solutions is not greater than $\sigma$.