Compactness, interpolation inequalities for small Lebesgue-Sobolev spaces and applications

We study some generalized small Lebesgue spaces and their associated Sobolev spaces. In particular, we prove that small Lebesgue-Sobolev spaces W1,(p(Ω) are compactly embedded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\frac{np}{n-p}}(\O),\ p<n$$\end{document}, p < n. As an application, we study variational problems involving critical exponents under multiple constraints.