Existence of minimizers for a quasilinear elliptic system of gradient type

Abstract

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

\begin{document}$ \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. $\end{document}

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

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

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.