Extremals for Poincaré-Sobolev sharp constants in Steiner symmetric sets
Lorenzo Brasco, Luca Briani, Francesca Prinari
TL;DR
We prove that open Steiner symmetric sets $\Omega\subset \mathbb{R}^N$, possibly unbounded, admit minimizers for the sharp Poincaré-Sobolev constant $\lambda_{p,q}(\Omega)$ and that the corresponding extremals solve the Lane-Emden equation $-\Delta_p u=\lambda_{p,q}(\Omega)|u|^{q-2}u$. Existence is obtained via an elementary Direct Method using a vanishing confinement perturbation and Steiner symmetrization to regain compactness, together with a uniform Harnack principle and Brézis–Lieb arguments to pass to the limit; minimizers exhibit exponential decay with rates determined by the underlying geometry, notably the inradius $r_\Omega$. A sharp geometric lower bound $\lambda_{p,q}(\Omega)\ge c(1/r_\Omega)^{N-p-\frac{p}{q}N}$ is established, and the results are illustrated with concrete examples and counterexamples highlighting the necessity of Steiner symmetry and convexity assumptions. Additionally, decay at infinity is developed for general nonnegative Lane-Emden subsolutions, with a Steiner-symmetric refinement yielding geometry-dependent, uniform decay constants for extremals.
Abstract
We prove existence of minimizers for the sharp Poincaré-Sobolev constant in general Steiner symmetric sets, in the subcritical and superhomogeneous regime. The sets considered are not necessarily bounded, thus the relevant embeddings may suffer from a lack of compactness. We prove existence by means of an elementary compactness method. We also prove an exponential decay at infinity for minimizers, showing that in the case of Steiner symmetric sets the relevant estimates only depend on the underlying geometry. Finally, we illustrate the optimality of the existence result, by means of some examples.