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

Dolbeault, Jean; Kowalczyk, Michał

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 949-977.

Cet article est consacré à la conjecture de Lin–Ni pour une équation semi-linéaire elliptique avec non-linéarité super-linéaire, sous-critique et des conditions de Neumann homogènes. Nous établissons un résultat de rigidité, c’est-à-dire nous prouvons que la seule solution positive est constante si le paramètre du problème est en dessous d’une borne explicite, reliée à la constante optimale d’une inégalité d’interpolation de Gagliardo–Nirenberg–Sobolev et aussi à une inégalité de Keller–Lieb–Thirring optimale. Nos résultats sont également valides dans un régime sous-linéaire. La borne de rigidité est obtenue par des méthodes de flots non-linéaires inspirées de résultats récents sur les variétés compactes, qui unifient des techniques d’équations elliptiques non-linéaires et la méthode du carré du champ en théorie des semi-groupes. Notre méthode requiert la convexité du domaine. Elle repose sur des quantités intégrales, prend en compte des estimations spectrales et fournit des inégalités améliorées.

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 carré 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.

Classification : 35J60, 26D10, 46E35
Keywords: semilinear elliptic equations, Lin–Ni conjecture, Sobolev inequality, interpolation, Gagliardo–Nirenberg inequalities, Keller–Lieb–Thirring inequality, optimal constants, rigidity results, uniqueness, carré du champ method, CD($\rho $, $N$) condition, bifurcation, multiplicity, generalized entropy methods, heat flow, nonlinear diffusion, spectral gap inequality, Poincaré inequality, improved inequalities, non-Lipschitz nonlinearity, compact support principle
Mot clés : Inégalités fonctionnelles

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