PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 9, Pages 2549–2557 S 0002-9939(05)07290-4 Article electronically published on April 12, 2005 UNIQUENESS OF POSITIVE SOLUTIONS FOR SINGULAR PROBLEMS INVOLVING THE p-LAPLACIAN ARKADY POLIAKOVSKY AND ITAI SHAFRIR (Communicated by David S. Tartakoff) Abstract. We study existence and uniqueness of positive eigenfunctions for the singular eigenvalue problem: −∆ p u − λη(x) u p-1 |x| p = μ u p-1 |x| p on a bounded smooth domain Ω ⊂ R N with zero boundary condition. We also characterize all positive solutions of −∆ p u = | N−p p | pu p-1 |x| p in R N \{0}. 1. Introduction This note is concerned with solutions of the following nonlinear eigenvalue prob- lem: (1.1)      −∆ p u − λη(x) |u| p-2 u |x| p = µ |u| p-2 u |x| p in Ω \{0}, u > 0 in Ω \{0}, u =0 on ∂Ω. Here Ω is a bounded domain in R N of class C 2 with 0 ∈ Ω, p ∈ (1, ∞) \{N }, η ∈ C α ( Ω) (α ∈ (0, 1)) such that η ≥ 0,η ≡ 0 in Ω and η(0) = 0, µ, λ ∈ R are two parameters and ∆ p u = div(|∇u| p−2 ∇u). By the regularity results of Tolksdorf [11] and Di Benedetto [4] any weak solution u ∈  r>0 W 1,p (Ω \ B r (0)) to (1.1) actually belongs to C 1 ( Ω \{0}). Naturally related to (1.1) is the following variational problem: (1.2) I [λ] := inf 0=v∈W 1,p 0 (Ω\{0})  Ω |∇v| p − λ  Ω η|v| p |x| p  Ω (|v|/|x|) p . Recall that in the case λ = 0 it follows from the well-known Hardy inequality, (1.3)  Ω |∇u| p ≥   N − p p   p  Ω (|u|/|x|) p , ∀u ∈ W 1,p 0 (Ω \{0}), that I [0] = c ∗ p,N := | N−p p | p . Moreover, by a simple construction of test functions approximating |x| 1−N/p (see [8]), it can be shown that we always have I [λ] ≤ c ∗ p,N . It was also proved in [8] (in the spirit of [3]) that there exists λ ∗ > 0 such that I [λ]= c ∗ p,N for λ ≤ λ ∗ and I [λ] <c ∗ p,N for λ>λ ∗ . If u ∈ W 1,p 0 (Ω \{0}) is a minimizer for (1.2), then u is clearly a solution to the equation in (1.1) with Received by the editors March 2, 2002. 2000 Mathematics Subject Classification. Primary 35J70; Secondary 49R50. c 2005 American Mathematical Society Reverts to public domain 28 years from publication 2549 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use