On the Semilinear Elliptic Equations b p g q n Duq 1q

In this paper, we consider the semilinear elliptic equation b g Duq upy uq in Rn, 1.1. m n 1q< x<. 1q< x<. where nG3, Ds nis1 ­2r­ xi2., b and g are two positive constants, and p, q, m, n are constants with q)p)1 and m Gn )2. We note that if b s0, g )0, and n )2, then the complete classification of all possible positive solutions was conducted by Cheng and Ni w Indiana Uni¨. Math. J. 41 1992., 261]278x. If g s0 and b )0, then 1.1. is the so-called Matukuma-type equation, and the solution structures were classified by Li and Ni w Duke Math. J. 53 1985., 895]924x and Ni and Yotsutani w Japan J. Appl. Math. 5 1988., 1]32x. If b )0 and g )0, then some results about the structure of positive solutions of 1.1.were derived by the first author wNonlinear Analysis, TM& A 28 1997., 1741]1750x. 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..