Existence of minimizers for some quasilinear elliptic problems

Abstract

The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem \begin{document}$ \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. $\end{document} 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.