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

Abstract

We consider the quasilinear elliptic equation,

$$ - {\text{div }}\left( {|\nabla u|^{m - 2} \nabla u} \right) = u^p + \lambda |u|^{m - 2} u$$

in 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 Pohožaev-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

$$\left( {\text{i}} \right) B {\text{is a ball, }}\lambda \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } {\text{0,1 < }}p + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \frac{{mn}}{{n - m}}{\text{for }}m < n {\text{and 1 < }}p < \infty {\text{for }}m = n, {\text{or}}$$

$$\left( {{\text{ii}}} \right) B {\text{is an annulus, }}\lambda \in \mathbb{R} {\text{and }}p = \frac{{mn}}{{n - m}}$$