On certain one-dimensional elliptic systems under different growth conditions at respective infinities

Abstract

A one-dimensional semi-linear elliptic system with constraint to the growth of its solutions at respective infinities is discussed. This system appears in a rescaling limit of a competition-diffusion system which describes very strong inter-specific competition between two biological species. A necessary and sufficient condition for the existence of solutions is characterized in terms of the constraint at respective infinities. Also the uniqueness of a solution and several asymptotic estimates of its derivatives at infinities are stated. Moreover similar problems are discussed for an associated inhomogeneous linear elliptic system with constraint to the growth of its solutions at infinities.