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 $$\bar{J}(u)=\frac{1}{p}{\int }_{\mathrm{\Omega }}\bar{A}(x,u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p}dx-{\int }_{\mathrm{\Omega }}G(x,u)dx$$ in the Banach space $X=W_0^{1,p}(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$, where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an open bounded domain, $1<p<N$ and the real terms $\bar{A}(x,t)$ and $G(x,t)$ are $C^1$ Carath'eodory functions on $\mathrm{\Omega }\times \mathbb{R}$. We prove that, even if the coefficient $\bar{A}(x,t)$ 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. The proof, which exploits the interaction between two different norms on $X$, is based on a weak version of the Cerami-Palais-Smale condition and a suitable intersection lemma which allow us to use a Mountain Pass Theorem.