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

Abstract

to appear in Archive for Rational Mechanics and AnalysisWe 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

A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

“…And indeed, the study of the system (4) becomes more delicate due to the existence of semi-trivial solutions, especially in the applications of Liouville-type theorems (cf. [6,19,16] and see Remark 3.1 below).…”

“…• Under the assumption m = 2, β ii = 0 for i = 1, 2 and β 12 > 0, the Liouvilletype theorem for positive solutions of problem (1) can be shown via comparison technique (see [28] and [31,19] for the elliptic case). More precisely, by taking the difference of the two equations and suitably using the maximum principle, we may show that u = v and thus reduce the system to a scalar equation.…”

Stationary States of Reaction-Diffusion and Schrödinger Systems with Inhomogeneous or Controlled Diffusion

“…This paper is a sequel to our recent work [28], in which we studied stationary states of systems of reaction-diffusion PDEs or standing waves of coupled Schrödinger systems, including as a particular case the important for applications system…”

“…We refer to Section 1.3 in [28] for a more detailed discussion on these applications, as well as references. We observe the last condition in (3) means the reaction in the system dominates the absorption, so there is no conservation of mass in the timedependent version of (1).…”

“…We observe the last condition in (3) means the reaction in the system dominates the absorption, so there is no conservation of mass in the timedependent version of (1). In [28] we studied the classification and non-existence of positive solutions of (1) in unbounded domains, as well as the a priori bounds and existence results which can be inferred in smooth bounded domains. A major difficulty in the study of system (1) is that it is in general neither cooperative (or quasi-monotone), nor variational in the sense that its solutions cannot be written as critical points of some functional defined on a Banach space.…”