Proportionality of Components, Liouville Theorems and a Priori Estimates for Noncooperative Elliptic Systems

Alexandre Montaru (1) , Boyan Sirakov (2) , Philippe Souplet (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Departamento de Matematica

We study qualitative properties of positive solutions of noncooperative, possibly nonvariational, elliptic systems. We obtain new classification and Liouville type theorems in the whole Euclidean space, as well as in half-spaces, and deduce a priori estimates and existence of positive solutions for related Dirichlet problems. We significantly improve the known results for a large class of systems involving a balance between repulsive and attractive terms. This class contains systems arising in biological models of Lotka-Volterra type, in physical models of Bose-Einstein condensates and in models of chemical reactions.

Domaines: Equations aux dérivées partielles [math.AP]

Alexandre Montaru, Boyan Sirakov, Philippe Souplet. Proportionality of Components, Liouville Theorems and a Priori Estimates for Noncooperative Elliptic Systems. Archive for Rational Mechanics and Analysis, 2014, 213, pp.129 - 169. ⟨10.1007/s00205-013-0719-4⟩. ⟨hal-00944637v2⟩