Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space

Montaru, Alexandre; Souplet, Philippe

Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 321-325.

Nous étudions les solutions classiques de systèmes elliptiques dans le demi-espace et donnons des conditions suffisantes assurant la symétrie (ou la proportionnalité) des composantes, i.e. $u=Kv$ avec $K>0$, ce qui réduit alors le système au cas scalaire. Sous une condition naturelle de structure sur les non-linéarités, nous montrons que les solutions à croissance sous-linéaire, donc en particulier les solutions bornées, sont symétriques. Ce résultat couvre le cas de systèmes non coopératifs, non variationnels et éventuellement sur-critiques. Nous obtenons aussi des résultats de proportionnalité sans hypothèse de croissance sur les solutions. Comme conséquence, nous obtenons de nouveaux théorèmes de type Liouville dans le demi-espace, ainsi que des estimations a priori et des résultats d'existence pour des problèmes de Dirichlet associés. Nos preuves reposent sur un principe du maximum, sur les propriétés de moyennes semi-sphériques, sur un résultat de rigidité pour les fonctions surharmoniques et sur la nonexistence de solution pour des inéquations scalaires dans le demi-espace.

We study classical solutions of elliptic systems in the half-space and provide sufficient conditions for having symmetry (or proportionality) of components, i.e. $u=Kv$ with $K>0$, which then reduces the system to the scalar case. Under a natural structure condition on the nonlinearities, we show that solutions with sublinear growth, hence in particular bounded solutions, are symmetric. Noncooperative, nonvariational systems as well as supercritical nonlinearities can be covered. We also give an instance of our proportionality results without growth restriction on the solutions. As a consequence, we obtain new Liouville-type theorems in the half-space, as well as a priori estimates and existence results for related Dirichlet problems. Our proofs are based on a maximum principle, on the properties of suitable half-spherical means, on a rigidity result for superharmonic functions and on nonexistence of solution for scalar inequalities on the half-space.

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