Minimization problem associated with an improved Hardy–Sobolev type inequality

Abstract

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): Ia≔infu∈W01,p(BR)∖{0}∫BR|∇u|pdx∫BR|u|p∗(s)Va(x)dxpp∗(s),whereVa(x)=1|x|s1−a|x|RN−pp−1β≥1|x|s, where 1<p<N and 0≤s≤p. The minimization problem Ia is equivalent to the minimization problem associated with the classical Hardy–Sobolev inequality on RN 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.