Abstract:
We prove that the least energy solution of the boundary value problem
$$
\begin{cases}
-\Delta u+u=|u|^{q-2}u&\text{ in }Q
\\
\frac{\partial u}{\partial\mathbf n}=0&\text{ on }\partial Q
\end{cases}
$$
is a constant for all $q\in(2;2^*]$ if $Q\subset\mathbb R^n$ ($n\ge 3$) is a sufficiently thin cylinder.