A sharp Hardy–Sobolev inequality with boundary term and applications

In this paper, we state a Hardy–Sobolev type inequality with boundary terms in a borderline case. As an application, we investigate the existence of solutions for a class of zero-mass quasilinear elliptic problem of the form -div(a(x)|∇u|N-2∇u)=k(x)f(u)inΩ,a(x)|∇u|N-2∇u·ν+|u|N-2u=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{rcllr} -\text {div}(a(x)|\nabla u|^{N-2}\nabla u) = k(x)f(u) &{}\text{ in } \Omega ,\\ a(x)|\nabla u|^{N-2}\left( \nabla u\cdot \nu \right) +|u|^{N-2}u=0 &{}\text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$\end{document}where Ω⊂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}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 2$$\end{document}, is an exterior domain, the weight functions a, k satisfy some growth conditions and the nonlinearity f has critical exponential growth.