Fractional Hardy-Sobolev equations with nonhomogeneous terms

This paper deals with existence and multiplicity of positive solutions to a class of nonlocal equations in the entire space, involving critical nonlinearities and Hardy potentials. We establish the profile decomposition of the Palais-Smale sequences associated with the functional. The existence proof in the nonlocal case and in the entire space is highly nontrivial and requires several delicate estimates and techniques to deal with. To the best of our knowledge, so far there has been no papers in the literature, where existence and multiplicity of positive entire solutions of Hardy-Sobolev type equations have been established in the nonhomogeneous case. Also the profile decomposition in the nonlocal case with the Hardy term is completely new and the proof is very involved, delicate and complicated compared with the local case s=1. The proofs are not at all an easy adaption of the local case or of the homogeneous case. The multiplicity results in this paper is new even in the local case s=1, but we leave the obvious changes, when s=1, to the interested reader.