Diffusion Growth Model:Equilibrium Solution Of Single Species

In the last 30 years,the research of reaction-diffusion equation with Dirichlet boundary condition is popular.And most has only one parameter m(x),which serves as both the intrinsic growth rate coefficient and carrying capacity of the population.In order to make the model more relevant for ecologists,we consider a logistic reac-tion term,with two parameters,r(x)for intrinsic growth rate,and K(x)for carrying capacity.The purpose of this paper is to investigate the existence and uniqueness of the positive steady state ud of (?) namely,the solution of the following equation (?) where u(x)represents the population density of a species at location x??;the migration rate d is assumed to be a positive constant;the habitat of the species ? is a bounded region in RN with smooth boundary (?)?;K(x)? C2(?)represents carrying capacity;r(x)?C2(?)represents intrinsic growth rate.We also study that as d? 0,ud?K in L??H1.The classical upper and lower solutions method is mainly method we use in this paper.