Multiple solutions for some symmetric supercritical problems

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem [ ar J(u) = rac1p int_Omega ar A(x,u)| abla u|^p dx - int_Omega G(x,u) dx ] in the Banach space $X = W^{1,p}_0(Omega)cap L^infty(Omega)$, where $Omega subset R^N$ is an open bounded domain, $1 < p < N$ and the real terms $ar A(x,t)$ and $G(x,t)$ are $C^1$ Carath'eo-do-ry functions on $Omega imes R$. We prove that, even if coefficient $ar A(x,t)$ makes the variational approach more difficult, if it satisfies ``good'' growth assumptions then at least one critical point exists also when the nonlinear term $G(x,t)$ has a suitable supercritical growth. Moreover, if the functional is even, it has infinitely many critical levels. We use variants of the classical Ambrosetti--Rabinowitz theorems which are based on a weak version of the Cerami--Palais--Smale condition.