On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

This paper deals with existence and multiplicity of positive solutions to the nonhomogeneous nonlocal equations with critical nonlinearity in the entire space. Hence the equations present a double loss of compactness which produces new interesting difficulties. To the best of our knowledge, so far there has been no papers in the literature, where existence and multiplicity of positive solutions of fractional Laplace equations, with the critical exponents in the entire space have been established in the nonhomogeneous case. The results in this paper are new even in the local case, but we leave the obvious changes to the interested reader. Even if the proof arguments rely on standard variational methods, the extension is not easy at all. Indeed, there are several difficulties to overcome, included the proof of the fact that the two constructed solutions are nontrivial, different and independent, as well as the profile decomposition.