Multiplicity results for some quasilinear equations in lack of symmetry

In this paper we prove the existence of multiple nontrivial solutions for the quasilinear equation in divergence form \[ - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = \lambda b(x,u) - g(x,u) \;\hbox{in $\Omega$,}\quad u = 0\; \hbox{on $\partial\Omega$,} \] in an open bounded domain $\Omega \subset \R^N$, where $A :\Omega \times \R \times \R^N \to \R$ is a given Carathéodory function with partial derivatives $A_t(x,t,\xi) = \frac{\partial A}{\partial t}(x,t,\xi)$ and $a(x,t,\xi) = (\frac{\partial A}{\partial \xi_1}(x,t,\xi),\dots,\frac{\partial A}{\partial \xi_N}(x,t,\xi))$. It generalizes the $p$-Laplacian problem \[ - \Delta_p u = \lambda |u|^{p-2}u - g(x,u), \qquad u \in W^{1,p}_0(\Omega), \] but, in general, the corresponding functional is not well defined in all the space $W^{1,p}_0(\Omega)$. Anyway, under suitable assumptions and by using variational tools, we are able to prove that the number of solutions of $(P_\lambda)$ depends on the parameter $\lambda$ and, even in lack of symmetry, at least three nontrivial solutions exist if $\lambda$ is large enough.