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} - {\rm{div}} (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 \mathbb 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.