In this paper, we consider the semilinear elliptic equationΔu+β1+|x|μup-γ1+|x|νuqinRn,where n≥3, Δ=∑ni=1(∂2/∂x2i), 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).
ASJC Scopus subject areas
- Applied Mathematics