Radial solutions of an elliptic equation with singular nonlinearity

For the equation − Δ u + u − β = u p , u > 0 in B R , u = 0 on ∂ B R , where B R ⊆ R N , 0 < β < 1 and 1 < p < N + 2 N − 2 if N ⩾ 3 , 1 < p < + ∞ if N = 2 , we show that there is R ¯ > 0 such that a radial solution u R exists if and only if 0 < R ⩽ R ¯ . It is unique in the class of radial solutions and u R ′ ( R ) < 0 if R < R ¯ , while u R ¯ ′ ( R ¯ ) = 0 . We also give a variational characterization of u R ¯ .