Singular elliptic problems in [formula omitted] with critical Sobolev–Hardy exponents

Abstract

Let N ≥ 3 , λ > 0 , β ≤ 0 , N + β − 2 > 0 , N + α > 0 , N + σ > 0 , α + 2 > β , σ + 2 > β , β 2 ≥ α p ( β , α ) , β 2 ≥ σ p ( β , σ ) , 1 < q < min ( p ( β , α ) , p ( β , σ ) ) , p ( s , t ) ≔ 2 ( N + t ) N + s − 2 be the critical Sobolev–Hardy exponent. Via the variational methods, we prove the existence of a nontrivial solution to the singular semilinear problem − div ( | x | β ∇ u ) = | x | α | u | p ( β , α ) − 2 u + λ a ( x ) | u | q − 2 u , u ≥ 0 in R N for suitable parameters N , λ , q and some kinds of functions a ( x ) .