IMPROVED SOBOLEV INEQUALITIES INVOLVING WEIGHTED MORREY NORMS AND THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO DOUBLY CRITICAL ELLIPTIC SYSTEMS INVOLVING FRACTIONAL LAPLACIAN AND HARDY TERMS

摘要

In this paper, we prove two new improved Sobolev inequalities involvingweighted Morrey norms in H~s(R~n)×H~s(R~n) and D~(1,p)(R~n)×D~(1,p)(R~n).For instance, the corresponding inequality in H~s(R~n)×H~s(R~n) states that:there exists C = C(n, s, α, η1， η2) > 0 such that for each (u， v) ∈H~s(R~n)×H~s(R~n), p∈[2，2_s~*(α)) and θ∈(θ,(2_(η1)/(2_s~*(α) ), it holds that(∫_(R~n)(|u|~(η1)|v|~(η2))/(|y|~α)dy~(1/(2_s~*(α)))≤C‖u‖_(H~s(R~n))~(θ/2)‖v‖(H~s(R~n))~(θ/2+(η2-η1)/(2_s~*(α)))‖(uv)‖_(L~(p/2,p/2(n-2s+r))(R~n,|y|~(-p/2r)~((η1)/(2_s~*(α))-θ/2)where s∈(0,1), 0<α<2s