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In this paper we consider the following coupled gradient-type quasilinearelliptic system \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x,u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &\hbox{ in$\Omega$,}\\[10pt] - {\rm div} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) =G_v\left(x, u, v\right) &\hbox{ in $\Omega$,}\\[10pt] u = v = 0 &\hbox{ on$\partial\Omega$,} \end{array} \right. \end{equation*} where $\Omega$ is anopen bounded domain in $\mathbb{R}^N$, $N\ge 2$. We suppose that some$\mathcal{C}^{1}$-Carath\'eodory functions $A,B:\Omega\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}$ exist suchthat $a(x,t,\xi) = \nabla_{\xi} A(x,t,\xi)$, $A_t(x,t,\xi) = \frac{\partialA}{\partial t} (x,t,\xi)$, $b(x,t,\xi) = \nabla_{\xi} B(x,t,\xi)$,$B_t(x,t,\xi) =\frac{\partial B}{\partial t}(x,t,\xi)$, and that $G_u(x, u,v)$, $G_v(x, u, v)$ are the partial derivatives of a$\mathcal{C}^{1}$-Carath\'eodory nonlinearity$G:\Omega\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$. Roughlyspeaking, we assume that $A(x,t,\xi)$ grows at least as$(1+|t|^{s_1p_1})|\xi|^{p_1}$, $p_1 > 1$, $s_1 \ge 0$, while $B(x,t,\xi)$ growsas $(1+|t|^{s_2p_2})|\xi|^{p_2}$, $p_2 > 1$, $s_2 \ge 0$, and that $G(x, u, v)$can also have a supercritical growth related to $s_1$ and $s_2$. Since thecoefficients depend on the solution and its gradient themselves, the study ofthe interaction of two different norms in a suitable Banach space is needed. Inspite of these difficulties, a variational approach is used to show that thesystem admits a nontrivial weak bounded solution and, under hypotheses ofsymmetry, infinitely many ones.

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