The authors deal with a three critical points theorem for a so-called variational-hemivariational inequality \[ \Phi^0(u,v-u)+\Psi(v)-\Psi(u)\geq 0\quad\forall v\in X \] where \(X\) is a Banach space, \(\Phi\) a locally Lipschitz continuous function, \(\Psi\) a convex, proper and lower semicontinuous function, and \(\Phi^0(u,v)\) denotes the generalized directional derivative of \(\Phi\) at \(u\) along \(v\). They obtain existence of multiple solutions for a Neumann elliptic variational-hemivariational inequality involving the \(p\)-Laplacian. Moreover, they study a Neumann problem for elliptic equations with discontinuous nonlinearities.