Minimization problem associated with an improved Hardy–Sobolev type inequality

We consider the existence or non-existence of a minimizer of the following minimization problems associated with an improved Hardy–Sobolev type inequality introduced by Ioku (2019): I a ≔ inf u ∈ W 0 1 , p ( B R ) ∖ { 0 } ∫ B R | ∇ u | p d x ∫ B R | u | p ∗ ( s ) V a ( x ) d x p p ∗ ( s ) , where V a ( x ) = 1 | x | s 1 − a | x | R N − p p − 1 β ≥ 1 | x | s , where 1 p N and 0 ≤ s ≤ p . The minimization problem I a is equivalent to the minimization problem associated with the classical Hardy–Sobolev inequality on R N via a transformation if we restrict ourselves to radial functions. In contrast to the classical results for a = 0 , we show the existence of non-radial minimizers for the Hardy–Sobolev critical exponent p ∗ ( s ) = p ( N − s ) N − p on bounded domains. Furthermore, as an application of the transformation, we derive an infinite-dimensional form of the classical Sobolev inequality in some sense.