The limiting behavior of constrained minimizers in Orlicz-Sobolev spaces

Let Ω ⊂ ℝN be a smooth, bounded domain. For each p > 1, let W01,Φp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_0^{1,{\Phi _p}}\left(\Omega \right)$$\end{document} be the Orlicz—Sobolev space generated by an N-function Φp and let up∈W01,Φp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_p} \in W_0^{1,{\Phi _p}}\left(\Omega \right)$$\end{document} be a nonnegative minimizer of the modular functional ∫ΩΦp(|∇u|)dx restricted to the sphere ∥u∥Lr(Ω) = 1, where r ≥ 1. Under certain hypotheses on the family (Φp), we prove that up converges uniformly to δ/∥δ∥Lr(Ω), where δ denotes the distance function to the boundary of Ω as p → ∞.