Multiple Solutions to a Transmission Problem with a Critical Hardy-Sobolev Exponential Source Term

In the paper there are established many results for a transmission problem with critical Hardy-Sobolev exponential source term u3|x|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{u<^>3}{|x|}$$\end{document} in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>3$$\end{document}. We obtain that there are at least three weakly nontrivial solutions when a positive coefficient of nonhomogeneous term is enough small using the variational method. Next infinitely many classical solutions are obtained when the coefficient equals to zero. Moreover, a new compactness condition is derived with the help of Brezis-Lieb's lemma and Mazur's lemma.