Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

In this paper we consider a coupled gradient-type quasilinear elliptic system stated in an open bounded domain in $R^N$, $Ngeq 2$. We suppose that the principal parts $A(x,t,\xi)$ and $B(x,t,\xi)$ grow at least as $(1+|t|^{s_1p_1})|\xi|^{p_1}$, $p_1 > 1$, $s_1 ge 0$, respectively as $(1+|t|^{s_2p_2})|\xi|^{p_2}$, $p_2 > 1$, $s_2 ge 0$, and that the nonlinear term $G(x, u, v)$ can also have a supercritical growth related to $s_1$ and $s_2$. Since the coefficients depend on the solution and its grandient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.