Analytical approach of the symmetry: Sharp supercritical Hardy–Sobolev inequalities and applications

Abstract

We consider the optimal Hardy–Sobolevinequality on smooth bounded symmetric domains of the Euclidean space without any assumption concerning the “shape” of the boundary (i.e. some convexity) confirming that the symmetry of a domain is an intrinsic property characterizing both the domain itself and its boundary. Our model domain is the solid torus because of its particular interest in terms both of the geometry and of the analysis. We apply the results of the above study to solve the supercritical problem (P)Δpu+a(x)up−1=f(x)up∗(s)−1|x|s,u>0onT,u=0on∂T,1<p<2,0≤s≤pandp∗(s)=p(2−s)2−pand some variants of it.