2019/08/11 by Sano, Megumi
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.03915
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. Ia := inf_u ∈ W01,p(BR ) ∖ \ 0\ \frac∫BR |∇ u |p dx( ∫BR |u|p^*(s) Va(x) dx )(p)/(p^*(s)), where Va (x) =\frac1|x|s ( 1- a ( (|x|)/(R) )(N-p)/(p-1) )β ≥ (1)/(|x|s). Only for radial functions, the minimization problem Ia is equivalent to it associated with the classical Hardy-Sobolev inequality on ℝN 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 Ia on balls BR. 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. 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.