Best constant in weighted Sobolev inequality with weights being powers of distance from the origin
The paper is devoted to the study of the best constant in the imbedding theorems for weighted Sobolev spaces with weight functions being powers of the distance from the origin in \(\mathbb{R}^n\). For instance, the author considers the variational problem: \[ S(p,q,\alpha,\beta,n)=\inf \Biggl[\int_{\mathbb{R}^n}|\nabla u|^p| x|^{p\alpha} dx : u \in W_{\alpha,\beta}^{1,p}(\mathbb{R}^n),\| u\|_{L_\beta^q}= 1\Biggr]. \] The invariance of \(\mathbb{R}^n\) by the group of dilatations may cause the loss of compactness, so the existence of extremals essentially depends on parameters. The author proves that under suitable relations between the parameters the extremals exist and this permits to get the best constant in the Sobolev inequality: \[ \int_{\mathbb{R}^n}|\nabla u|^p| x|^{p\alpha} dx\geq S(p,q,\alpha,\beta,n) \Biggl(\int_{\mathbb{R}^n}| u|^q| x|^{q\beta} dx\Biggr)^{p/q}, \] where \(u\) is any function in \(C_0^1(\mathbb{R}^n)\). The obtained theorems generalize some earlier results of Talenti, Maz'ya and Egnell.
scientific article; zbMATH DE number 1139490