Asymptotic Behavior And Stability Of Positive Solutions Of Non-linear Partial Differential Equations

In this dissertation, we will consider the asymptotic behavior and the stability of the positive solutions of the following equationwhich are positive steady states of the following Cauchy problem:where p>1,x∈Rⁿ,n≥3,â–³=(?) is the n-dimensional Laplacian, T>0 is some positive constant.The equation arisen in Riemannian geometry, is said to be the conformal scalar curvature equations. A simple version isFor the physical reasons and because of the results on the symmetry of positive solutions, many mathematicians study the radial solutions, such as the existence of positive radial solutions and asymptotic behavior at infinity etc.The underlying obstacles result from the presence of the singularity in the origin of the second terms F(x, u), inhomogeneity and the lack of compactness due to the unbounded domain (entire spaces). This dissertation deserve to the radial solutions, with r=|x|, the equation (1) reduce to the followingDenote the positive solution of the equation (3) by u_α(r) with initial value u(0)=α.For everyα>0, there is a positive solution u_α(r) for equation (0.0.3) with some assumptions on F(x, u). We will study the asymptotic behavior and the semistability of the minimal positive steady state of equation (1) for F(x, u)= K(x)u^p+μf(x). In addition, we will prove that all slow decay positive steady states of equation (1) are stable and weakly asymptotically stable in some weighted L^∞norms. When F(x,u) = |x|^l₁u^p+|x|^l₂u^q, we will consider the asymptotic behavior of positive solution of (3) near infinity.