On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents

Suppose . . .N(N . 3) is a smooth bounded domain, .i..,0<ai<.....,..:=(N.22)2,0..i<(......ai)2,ai<bi<ai+1 and pi:=2NN.2(1+ai.bi) are the weighted critical Hardy-Sobolev exponents, i = 1, 2,., k, k . 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem .i=1k(.div(|x..i|.2ai.u)..iu|x..i|2(1+ai).upi.1|x..i|bipi)=0 w ith Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters a i , b i and . i , i = 1, 2,., k.