Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities and spectral estimates

TL;DR

This paper proves a new rigidity result for semi-linear elliptic equations, showing that solutions are constant below a certain parameter threshold, using advanced inequalities and spectral estimates.

Contribution

It introduces a novel rigidity theorem linking solution uniqueness to optimal interpolation and spectral inequalities, employing nonlinear flow methods on convex domains.

Findings

Unique positive solutions are constant below an explicit parameter bound.

The bound relates to optimal Gagliardo-Nirenberg-Sobolev and Keller-Lieb-Thirring inequalities.

The method unifies nonlinear elliptic techniques with semi-group spectral methods.

Abstract

This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo-Nirenberg-Sobolev interpolation inequality and also with an optimal Keller-Lieb-Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow methods inspired by recent results on compact manifolds, which unify nonlinear elliptic techniques and the carr{\'e} du champ method in semi-group theory. Our method requires the convexity of the domain. It relies on integral quantities, takes into account spectral estimates and provides improved functional inequalities.