The aim of this talk is to present some recent results about the existence and the multiplicity of solutions of the elliptic problem $$(P)\qquad\qquad \left\{ \begin{array}{lll} \displaystyle{-\Delta_p u\ =\ g(x, u) + \varepsilon h(x, u)} & \mbox{ in } \Omega,\\ \displaystyle{u=0} & \mbox{ on } \partial\Omega,\\ \end{array} \right. $$ where $1\lt p\lt +\infty$, $\Delta_p u= {\rm div}(|\nabla u|^{p-2}\nabla u)$, $\Omega$ is an open bounded domain of ${\bf R}^N$ with smooth boundary $\partial\Omega$, $\varepsilon\in {\bf R}$, $g$ is subcritical and asymptotically $(p-1)$-linear at infinity and $h$ is just a continuous function. For $p=2$, even when this problem has not a variational structure on $W^{1,2}_0(\Omega)$, suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is "stable" under small perturbations, in particular obtaining multiplicity results if $g$ is odd, both in the non-resonant and in the resonant case. For $p\not=2$ and $\varepsilon=0$, we get the existence and the multiplicity of solutions of the quasilinear elliptic problem $(P)$ by means of some abstract critical point theorems on Banach spaces and using two sequences of quasi-eigenvalues for the $p$-Laplacian operator. Finally we use the so-called Bolle's method to get the existence of infinitely many solutions of $(P)$ when $g(x,u)=|u|^{q-2}$, $p \lt q \lt p^\ast$, $\varepsilon=1$, $h(x,u)=h(x)$ and $u=\varphi$ on $\partial\Omega$, with $\varphi\in C^2(\overline\Omega)$.