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

Abstract

We obtain classification, solvability, and nonexistence theorems for positive stationary states of reaction-diffusion and Schrödinger systems involving a balance between repulsive and attractive terms. This class of systems contains PDE arising in biological models of Lotka--Volterra type, in physical models of Bose--Einstein condensates, and in models of chemical reactions. We show, with different proofs, that the results obtained in [A. Montaru, B. Sirakov, and P. Souplet, Arch. Ration. Mech. Anal., 213 (2014), pp. 129--169] for models with homogeneous diffusion are valid for general heterogeneous media, and even for controlled inhomogeneous diffusions.