On the semilinear elliptic equations Delta u+beta/(1+vertical bar x vertical bar)(mu) u(p)-gamma/(1+vertical bar x vertical bar)(nu) u(q)=0 in R-n

摘要

In this paper, we consider the semilinear elliptic equation Delta u + beta/(1 + x )(mu) u(P) - gamma/(1 + x )(nu) u(q) in R-n, (1.1) where n greater than or equal to 3, Delta = Sigma(i-1)(n)(partial derivative(2)/partial derivative x(i)(2)), beta and gamma are two positive constants, and p, q, mu, nu are constants with q > p > 1 and mu greater than or equal to nu > 2. We note that if beta = 0, gamma > 0, and nu > 2, then the complete classification of all possible positive solutions was conducted by Cheng and Ni [Indiana Univ. Math. J. 41 (1992), 261-278]. If gamma = 0 and beta > 0, then (1.1) is the so-called Matukuma-type equation, and the solution structures were classified by Li and Ni [Duke Math. J. 53 (1985), 895-924] and Ni and Yotsutani [Japan J. Appl. Math. 5 (1988), 1-32]. If beta > 0 and gamma > 0, then some results about the structure of positive solutions of (1.1) were derived by the first author [Nonlinear Analysis, TM & A 28 (1997), 1741-1750]. The purpose of this paper is to discuss the uniqueness and properties of unbounded positive solutions and investigate some further structures of the positive solutions of Eq. (1.1). (C) 2000 Academic Press. [References: 12]