Extremal functions for the Caffarelli-Kohn-Nirenberg inequalities: A simple proof of the symmetry.(English)Zbl 1171.35041

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 $$P$$-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 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 [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)].

MSC:

 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations

Zbl 1036.35028
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References:

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