Solutions for quasilinear elliptic problems with critical Sobolev–Hardy exponents

Abstract

Let N ⩾ 3 , 2 < p < N , 0 ⩽ s < p and p * ( s ) : = p ( N − s ) N − p . Via the variational methods and analytic technique, we prove the existence of nontrivial solution to the singular quasilinear problem − Δ p u + | u | p − 2 u = | u | p * ( s ) − 2 | x | s u + f ( u ) , u ∈ W r 1 , p ( R N ) for N ⩾ p 2 and suitable functions f ( u ) .