On the Semilinear Elliptic Equations (formula presented)

In this paper, we consider the semilinear elliptic equationΔu+β1+|x|^μu^p-γ1+|x|^νu^qinRⁿ,where n≥3, Δ=∑ⁿ_i=1(∂²/∂x²_i), β and γ are two positive constants, and p,q,μ,ν are constants with qp1 and μ≥ν2. We note that if β=0, γ0, and ν2, then the complete classification of all possible positive solutions was conducted by Cheng and Ni [Indiana Univ. Math. J.41 (1992), 261-278]. If γ=0 and β0, 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 β0 and γ0, 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).