Uniqueness of positive radial solutions for semilinear elliptic equations on annular domains

Abstract

where 5 = 5a;b = {x ∈ Rn; 0iai |x|ib} is an annular domain in Rn; n ≥ 3. It is well known that the uniqueness problems of solutions of (1.1) and (1.2) are both fundamental and often di8cult. During the last decade, there has been tremendous progress in studying these problems when 5 is a ball in Rn or entire Rn n ≥ 3, see, e.g., [1,2,5,10,11]. However, very little was known concerning the annular domain until the work of Ni and Nussbaum [12]. Until very recently, Co<man [3] showed that when n=3; f(u)=−u+ up; 1ip ≤ 3, (1.1) and (1.2) have unique positive radial solution for any annular domain, see also [13]. Then, it is of interest to extend Co<man’s result to a more general situation. For example, we may ask the following question.