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

Consider the following Neumann problem

(*)

where d > 0, B ₁ is the unit ball in ℝ ^N , k(x) = k(|x|) ≢ 0 is nonnegative and in 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. Supported by the National Natural Science Foundation of China (No. 10571174, 10631030), Chinese Academy of Sciences grant KJCX3-SYW-S03.