Uniqueness of the least-energy solution for a semilinear Neumann problem

Grossi, Massimo

doi:10.1090/S0002-9939-99-05491-X

Uniqueness of the least-energy solution for a semilinear Neumann problem
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Proc. Amer. Math. Soc. 128 (2000), 1665-1672 Request permission

Abstract:

We prove that the least-energy solution of the problem \[ \begin {cases} -d\Delta u+u=u^p\quad & \text {in $B$},\\ u>0 & \text {in $B$},\\ \frac {\partial u}{\partial \nu } = 0 & \text {on $\partial B$}, \end {cases} \] where $B$ is a ball, $d>0$ and $1<p<{{N+2}\over {N-2}}$ if $N\ge 3$, $p>1$ if $N=2$, is unique (up to rotation) if $d$ is small enough.

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