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 \(\frac{u^3}{|x|}\) $\frac{u^3}{|x|}$ in \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ . 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.