Existence of minimizers for some quasilinear elliptic problems

The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem \[ \left\{\begin{array}{ll} - \divg (a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ f(x,u) &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} \right. \] where $\Omega \subset \R^N$ is an open bounded domain and $A(x,t,\xi)$, $f(x,t)$ are given real functions, with $A_t = \frac{\partial A}{\partial t}$, $a = \nabla_\xi A$. We prove that, even if $A(x,t,\xi)$ makes the variational approach more difficult, the functional associated to such a problem is bounded from below and attains its infimum when the growth of the nonlinear term $f(x,t)$ is ``controlled'' by $A(x,t,\xi)$. Moreover, stronger assumptions allow us to find the existence of at least one positive solution. We use a suitable Minimum Principle based on a weak version of the Cerami-Palais-Smale condition.