Weyl-type eigenvalue bounds for the fractional p-Laplacian

In this paper we study Weyl-type eigenvalue bounds for the variational eigenvalues of the fractional p-Laplacian on a bounded domain Omega subset of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset \mathbb{R}<^>{n}$\end{document} with a Lipschitz boundary. We prove that lambda m(Omega)>= C|Omega|-spnmspn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{m}(\Omega )\geq C |\Omega |<^>{-\frac{sp}{n}} m<^>{\frac{sp}{n}}$\end{document}, where C=C(s,p,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C=C(s,p,n)$\end{document}. This result partially confirms a conjecture of Iannizzotto and Squassina (Asymptot. Anal. 88: 233-245, 2014). We also study the auxiliary weighted eigenvalue problem (1.2) and obtain that lambda m(w,Omega)>= C(integral Omega wr(x)dx)-1r|Omega|1r-spnmspn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{m}(w, \Omega )\geq C\Bigl(\int _{\Omega }w<^>{r}(x)dx \Bigr)<^>{-\frac{1}{r}}| \Omega |<^>{\frac{1}{r}-\frac{sp}{n}} m<^>{\frac{sp}{n}}$\end{document}, where C=C(s,p,n,r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C=C(s,p,n,r)$\end{document}. The main method used in this paper is the approximation method developed by Birman and Solomyak (Quantitative Analysis in Sobolev Imbedding Theorems and Applications to Spectral Theory, 1980).