Minimization problem associated with an improved Hardy-Sobolev type inequality

Abstract

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. $$I_a:=\underset{u\in W_0^{1,p}(B_R)\setminus \{0\}}{\inf }\frac{{\int }_{B_R}|\nabla u{|}^{p}dx}{{\left({\int }_{B_R}|u{|}^{p^{\ast }(s)}{V}_a(x)dx\right)}^{\frac{p}{p^{\ast }(s)}}},\text{where}{V}_a(x)=\frac{1}{|x{|}^s{\left(1-a{\left(\frac{|x|}{R}\right)}^{\frac{N-p}{p-1}}\right)}^{\beta }}\ge \frac{1}{|x{|}^s}.$$ 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 $p^{\ast }(s)=\frac{p(N-s)}{N-p}$ 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.

24 pages