Existence of minimizers for a quasilinear elliptic system of gradient type

Abstract

<p style='text-indent:20px;'>The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb R^N $\end{document}</tex-math></inline-formula> is an open bounded domain, <inline-formula><tex-math id="M2">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ A(x,t,\xi) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ B(x,t, {\xi}) $\end{document}</tex-math></inline-formula> are <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{C}^1 $\end{document}</tex-math></inline-formula>–Carathéodory functions on <inline-formula><tex-math id="M6">\begin{document}$ \Omega \times \mathbb R \times { \mathbb R}^{N} $\end{document}</tex-math></inline-formula> with partial derivatives <inline-formula><tex-math id="M7">\begin{document}$ A_t = \frac{\partial A}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ a = {\nabla}_{\xi}A $\end{document}</tex-math></inline-formula>, respectively <inline-formula><tex-math id="M9">\begin{document}$ B_t = \frac{\partial B}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ b = {\nabla}_{{\xi}}B $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M11">\begin{document}$ g_1(x,t,s) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ g_2(x,t,s) $\end{document}</tex-math></inline-formula> are given Carathéodory maps defined on <inline-formula><tex-math id="M13">\begin{document}$ \Omega \times \mathbb R\times \mathbb R $\end{document}</tex-math></inline-formula> which are partial derivatives with respect to <inline-formula><tex-math id="M14">\begin{document}$ t $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ s $\end{document}</tex-math></inline-formula> of a function <inline-formula><tex-math id="M16">\begin{document}$ G(x,t,s) $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We prove that, even if the general form of the terms <inline-formula><tex-math id="M17">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$ B $\end{document}</tex-math></inline-formula> makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space <inline-formula><tex-math id="M19">\begin{document}$ X $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>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.</p>