A multiplicity theorem for the Neumann problem

Here is a particular case of the main result of this paper: Let $\Omega \subset {\mathbb{R}}^{n}$ be a bounded domain, with a boundary of class $C^{2}$, and let $f, g : {\mathbb{R}}\to {\mathbb{R}}$ be two continuous functions, $\alpha \in L^{\infty }(\Omega )$, with $\hbox{\rm ess inf}_{\Omega }\alpha >0$, $\beta \in L^{p}(\Omega )$, with $p>n$. If

and if the set of all global minima of the function $\xi \to {\frac{{\xi^{2}}}{{2}}}-\int _{0}^{\xi }f(t)\,dt$ has at least $k\ge 2$ connected components, then, for each $\lambda >0$ small enough, the Neumann problem

admits at least $k+1$ strong solutions in $W^{2,p}(\Omega )$.