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

Adimurthi, ; Yadava, S. L. (1994) An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem Archive for Rational Mechanics and Analysis, 127 (3). pp. 219-229. ISSN 0003-9527

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Related URL: http://dx.doi.org/10.1007/BF00381159

Abstract

We consider the quasilinear elliptic equation, - div (|∇u|m - 2u) = up + 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.

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