An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem

Abstract

We consider the quasilinear elliptic equation, - div (|∇_u|^{m - 2}∇_u) = u^p + λ|u|^{m - 2} u in B where B is a ball or an annulus in Rⁿ, 1 < m ≦ n, p is a positive real number, and λ ε R. Using a generalized Pohozaev-type variational identity of Ni & Serrin or Pucci and Serrin and an elementary calculus lemma, we establish uniqueness of positive radial solutions for the Dirichlet boundary condition if either (i) B is a ball, λ ≧ 0, 1 < p + 1 ≦ mn/n-m for m < n and 1 < p < ∞ for m = n, or (ii) B is an annulus, λ ∈ R and p = mn/ n-m.