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Archive for Rational Mechanics and Analysis

, Volume 142, Issue 2, pp 127–141 | Cite as

Uniqueness of Positive Solutions of Δu+f(u)=0 in ℝN, N≦3

  • Carmen Cortázar
  • Manuel Elgueta
  • Patricio Felmer
Article

Abstract.

We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation \(\Delta u+f(u)=0 \eqno{(*)}\) in ℝ N . We assume that the function f has two zeros, the origin and u 0>0. Above u 0 the function f is positive, is locally Lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature. Below u 0, f is assumed to be non-positive, non-identically zero and merely continuous. Our results are obtained through a careful analysis of the solutions of an associated initial‐value problem, and the use of a monotone separation theorem. It is known that, for a large class of functions f, the ground states of (*) are radially symmetric. In these cases our result implies that (*) possesses at most one ground state.

Keywords

Differential Equation Growth Condition Partial Differential Equation Large Class Careful Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Carmen Cortázar
    • 1
  • Manuel Elgueta
    • 2
  • Patricio Felmer
    • 2
  1. 1.Facultad de Matemáticas, Universidad Católica, Casilla 306 Correo 22, Santiago, ChileCL
  2. 2.Departamento de Ing. Matemática, F.C.F.M., Universidad de Chile, Casilla 170 Correo 3, Santiago, ChileCL

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