Extremal functions for the Caffarelli-Kohn-Nirenberg inequalities: A simple proof of the symmetry
Let \(\Omega\) denote an open subset of the cylinder \({\mathcal C}:=S^{N-1}\times{\mathbb R}\). The authors consider symmetry of solutions to the equation \[ -\Delta_\sigma v+\lambda v=f(v),\qquad \sigma\in\Omega,\tag{1} \] posed in \(H^1_0(\Omega)\), with \(\lambda\geq0\), \(f\) continuous, and \(f(0)=0\). Fixing \(P\in S^{N-1}\), \(\Omega\) is \textit{\(P\)-symmetric} if \(\Omega\cap({\mathbb R}^N\times\{t\})\) is a geodesic ball in \(S^{N-1}\times\{t\}\) with center \((P,t)\), for every \(t\in{\mathbb R}\). Suppose from now on that \(\Omega\) is \(P\)-symmetric. Informally, a measurable function \(v:\Omega\to{\mathbb R}\) is \textit{foliated Schwarz symmetric with respect to \(P\)} if there is \(g:[0,\pi]\times{\mathbb R}\to{\mathbb R}\) which is nonincreasing in its first argument, and such that \[ v(\theta,t)=g(\text{dist}(\theta,P),t) \] for every \((\theta,t)\in\Omega\). Here \(\text{dist}(\theta,P)\) denotes the geodesic distance of \(\theta\) and \(P\) on \(S^{N-1}\). Denote \(F(u):=\int_0^u f(s)\,ds\) and \[ \Phi(v):=\frac12\int_\Omega(|\nabla v|^2+v^2)\,d\sigma-\int_\Omega F(v)\,d\sigma, \] for \(v\in H^1_0(\Omega)\). It is proved that every solution \(v\) of (1) that minimizes \(\Phi\) among all nontrivial solutions is foliated Schwarz symmetric. The proof is very short and relies on a technique based on polarizations, as developed by one of the authors. It is then shown how this result implies foliated Schwarz symmetry for extremal functions in the Caffarelli-Kohn-Nirenberg inequalities, as previously shown in [\textit{C.-S. Lin, Z.-Q. Wang}, Proc. Am. Math. Soc. 132, No. 6, 1685--1691 (2004); erratum ibid. 132, No. 2183 (2004; Zbl 1036.35028)].
scientific article; zbMATH DE number 5543856