Uniqueness and radial symmetry of least energy solution for a semilinear Neumann problem

Abstract

Consider the following Neumann problem

$$ d\Delta u - u + k(x)u^p = 0 and u > 0 in B_1 , \frac{{\partial u}} {{\partial \nu }} = 0 on \partial B_1 , $$

(*)

where d > 0, B ₁ is the unit ball in ℝ^N, k(x) = k(|x|) ≢ 0 is nonnegative and in $C(\bar B_1 ), 1 < p < \frac{{N + 2}} {{N - 2}}$ with N ≥ 3. It was shown in [2] that, for any d > 0, problem (*) has no nonconstant radially symmetric least energy solution if k(x) ≡ 1. By an implicit function theorem we prove that there is d ₀ > 0 such that (*) has a unique radially symmetric least energy solution if d > d ₀, this solution is constant if k(x) ≡ 1 and nonconstant if k(x) ≢ 1. In particular, for k(x) ≡ 1, d ₀ can be expressed explicitly.