Nonlinear Eigenvalues for a Quasilinear Elliptic System in Orlicz–Sobolev Spaces

Using an Orlicz–Sobolev Space setting, we consider an eigenvalue problem for a system of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{l}{ - \Delta _{\Phi _1 } u = \lambda (a_1 (x,u) + b(x)\gamma _1 (u)\Gamma_2 (v))\quad \text{in}\;\Omega ,} \\{ - \Delta _{\Phi _2 } v = \lambda (a_2 (x,v) + b(x)\Gamma _1 (u)\gamma_2 (v))\quad \hbox{in}\;\Omega ,} \\{u = v = 0\quad \hbox{on}\;\partial \Omega .} \\ \end{array} } \right.$$\end{document}We prove that the solution to a suitable minimizing problem, with a restriction, yields a solution to this problem for a certain λ. The differential operators involved lack homogeneity and in addition the Orlicz–Sobolev spaces needed may not be reflexive and the corresponding functional in the minimization problem is in general neither everywhere defined nor a fortiori C1.