Uniqueness of positive radial solutions of a semilinear Dirichlet problem in an annulus

Abstract

We establish the uniqueness of positive radial solutions of -Δ u = u^ρ - u in B(R₁, R₂), u = 0 on ∂ B(R₁, R₂), where B(R₁, R₂) is an annulus and 0 < R₁ < R₂ ≤ ∞ , in the following cases. (a) n ∈ {3, 4} and 1 < p ≤ n/(n - 2). (b) n ∈ {5, 6, 7, 8} and 1 < p ≤ p₀(n) for some p₀(n) < n/(n - 2). Earlier to this result, the uniqueness has been obtained by Coffman for n = 3 and 1 < p ≤ 3 and by Yadava for p ≥ (n + 2)/(n - 2) and n ≥ 3.