Asymptotic Behaviors For Positive Solutions Of Some Nonlinear Elliptic Equation

In this paper we mainly study the asymptotic behavior of positive solutions of the following equationswhere -2＜1＜0,1＜p＜q,â–³= (?).The equations arised in Riemannian geometry, are called the conformal scalar curvature equations. The first equation after simplication isBecause of the physical background and because of the results on the symmetry of positive solutions , most of mathematicians study equations . such as the existence of positive radial solutions and asymptotic behavior at infinity etc. In 1992, M. K. Kwong, J. B. Mcleod, L. A. Peletier and W. C. Troy [19] have studied the equationin Rⁿ, n＞2. They obtained the existence and uniqueness of positive, radial sym-metric solution to the equation under the condition q＞p＞(n + 2)/(n - 2). We study the asymptotic behavior of positive radial solutions of the equationin Rⁿ. This equation arises in various problems in applied mathematics, e.g. in the study of phase transitions, nuclear cores and more recently in population genetics. This dissertation will study the radial solutions to equation (1), with r = |x|, the equations reduce to the following: In particular, we will study the asymptotic behavior of positive radial solutions of the nonlinear equation (1) at infinity, where p , q , l₁ , l₂, satisfying for some positive constants m, L, such thatAs for equation (2), we also study the asymptotic behavior at zero.