Regularity Of Solutions For Singular Elliptic Equation Involving Hardy-Sobolev-Maz’ya Term

In this paper,we consider the solution’s regularity of following PDE involving Hardy-Sobolev-Maz’ya terms where RN=Rk×RN-k,2≤k<N,N≥3,0≤α≤(k-2)/2,0≤t<2:d≤b≤d+1,(k-2-2a)2/4≤λ≤(k-2)2/4, pt=(N=2-2t)(N-2),A point x∈RN is denoted x=(y,z)∈Rk×RN-k and0∈ΩRN is a bounded domain with smooth boundary.In this article we mainly show the solution’s regularity of the problem.In order to prove the result,we transform the problem to a simpler equation. then when using the method of Moser iteration we can obtain the problem trans-formed,the solution of the problem belongs to Lloc∞(Ω,|y|-bp).After that we apply our theorem to the scalar curvature problem on the CR sphere,we can get the solution’s regularity of function一△Hn=Φ(ζ)u(ζ)(Q+2)/(Q-2),ζ∈Hn, that is u∈Lloc∞(Ω).At last,we discuss some more general Hardy-Sobolev,type inequalities in-volving the weighted Dirichlet integral.