On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems

We study a quasilinear elliptic problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \left\{\begin{array}{ll} -\,{\rm div}(\phi(|\nabla u|)\nabla u) = \lambda f(x,u) &\quad\text{in ${\rm \Omega}$} \\ u = 0 &\quad\text{on $\partial{\rm \Omega}$}\end{array}\right.$$ \end{document} with nonhomogeneous principal part φ. Under the hypothesis f(x,t)= o(φ(t)t) at t= 0 and ∞, the existence of multiple positive solutions is proved by using the variational arguments in the Orlicz–Sobolev spaces.