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On the uniqueness of solutions of a semilinear equation in an annulus

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    * Corresponding author 
This research was supported by FONDECYT-1190102 for the first and second author, and FONDECYT- 1170665 for the third author and by JSPS KAKENHI Grant Number 19K03595 and 17H01095 for the fourth author
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  • We establish the uniqueness of positive radial solutions of

    $ \begin{align} \begin{cases} \Delta u +f(u) = 0, \quad x\in A \\ u(x) = 0 \qquad \qquad x\in \partial A \end{cases} \;\;\;\; (P)\end{align} $

    where $ A: = A_{a, b} = \{ x\in {\mathbb R}^n : a<|x|<b \} $, $ 0<a<b\le\infty $. We assume that the nonlinearity $ f\in C[0, \infty)\cap C^1(0, \infty) $ is such that $ f(0) = 0 $ and satisfies some convexity and growth conditions, and either $ f(s)>0 $ for all $ s>0 $, or has one zero at $ B>0 $, is non positive and not identically 0 in $ (0, B) $ and it is positive in $ (B, \infty) $.

    Mathematics Subject Classification: Primary: 35J61, 35A02; Secondary: 35A24.


    \begin{equation} \\ \end{equation}
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