This paper is concerned with existence of non-trivial weak solutions for quasilinear elliptic equations of the form \(-\text{div} A(x,\nabla u)=f(x,u)\) in \(\Omega\) subject to homogeneous Neumann boundary conditions. Here \(\Omega\) is a smooth and bounded domain in \(\mathbb R^N\), \(A:\overline \Omega\times\mathbb R^N\rightarrow\mathbb R^N\) is a strictly monotone mapping with respect to the second variable which satisfies some general assumptions without being necessarily \((p-1)\)-homogeneous. The differential operator considered by the authors includes the \(p\)-Laplace case. Under some assumptions on the data \(f\), the existence of one, two or three non-trivial weak solutions is proved. Particular attention is paid to the case \(f(x,u)=\alpha u^{p-1}_+-\beta u^{p-1}_{-}\). The proofs rely on variational techniques which combine mountain pass methods, deformation lemma and minimax principle with minimization, truncation, regularity and linking.