The Neumann problem for semilinear elliptic equation in thin cylinder. The least energy solutions

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.