Existence of minimizers for a quasilinear elliptic system of gradient type

The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type

$ \left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right. $

where $ \Omega \subset \mathbb R^N $ is an open bounded domain, $ N \geq 2 $ and $ A(x,t,\xi) $, $ B(x,t, {\xi}) $ are $ \mathcal{C}^1 $–Carathéodory functions on $ \Omega \times \mathbb R \times { \mathbb R}^{N} $ with partial derivatives $ A_t = \frac{\partial A}{\partial t} $, $ a = {\nabla}_{\xi}A $, respectively $ B_t = \frac{\partial B}{\partial t} $, $ b = {\nabla}_{{\xi}}B $, while $ g_1(x,t,s) $, $ g_2(x,t,s) $ are given Carathéodory maps defined on $ \Omega \times \mathbb R\times \mathbb R $ which are partial derivatives with respect to $ t $ and $ s $ of a function $ G(x,t,s) $.

We prove that, even if the general form of the terms $ A $ and $ B $ makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space $ X $.

The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem.