Minimization problem associated with an improved Hardy-Sobolev type inequality

We consider the existence and the non-existence of a minimizer of the following minimization problems associated with an improved Hardy-Sobolev type inequality introduced by Ioku. $I_a := \inf_{u \in W_0^{1,p}(B_R ) \setminus \{ 0\} } \frac{\int_{B_R} |\nabla u |^{p} \,dx}{\left( \int_{B_R} |u|^{p^*(s)} V_a(x) \,dx \right)^{\frac{p}{p^*(s)}}}, \,\,\text{where}\,\, V_a (x) =\frac{1}{|x|^s \left( 1- a \,\left( \frac{|x|}{R} \right)^{\frac{N-p}{p-1}} \right)^β} \ge \frac{1}{|x|^s}.$ Only for radial functions, the minimization problem $I_a$ is equivalent to it associated with the classical Hardy-Sobolev inequality on $\mathbb{R}^N$ via a transformation. First, we summarize various transformations including that transformation and give a viewpoint of such transformations. As an application of this viewpoint, we derive {\it an infinite dimensional form} of the classical Sobolev inequality in some sense. Next, without the transformation, we investigate the minimization problems $I_a$ on balls $B_R$. In contrast to the classical results for $a=0$, we show the existence of non-radial minimizers for the Hardy-Sobolev critical exponent $p^* (s)=\frac{p (N-s)}{N-p}$ on bounded domains. Finally, we give remarks of a different structure between two nonlinear scalings which are equivalent to the usual scaling only for radial functions under some transformations.

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