A rearrangement based proof for the existence of extremal functions for the Sobolev-Poincaré inequality on \(B^n\)
Let \(B^n\) be the open ball in \({\mathbb R}^n\), \(n\geq 2\). Assume that \(p\) is a real number such that \(1<p\leq (n+1)/2\) and \(q=np/(n-p)\). Let \(W^{1,p}({\mathbb R}^n)\) be the completion of \(C^\infty (B^n)\) in the norm \(\int_{B^n}(|u|^p+|\nabla u|^p)\,dx<\infty\). Let \(\bar u\) denote the mean-value of \(u\) over \(B^n\). The main result of this paper asserts that, under these assumptions, there is an extremal function for \[ \lambda (n,p):=\inf\frac{\|\nabla u\|_p}{\| u-\bar u\|_q}>0. \] The proof is based on the following facts: (i) the concentrated compactness principle asserts that a maximizing sequence must either converge to an extremal or to a delta function at a point on the boundary; (ii) a maximizing sequence must either converge to an extremal or to a delta function at \(-1\) or \(1\).
scientific article; zbMATH DE number 5651224