Geodesic
The Neumann problem for semilinear elliptic equation in thin cylinder.
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 38, Tome 348 (2007), pp. 272-302

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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. 
@article{ZNSL_2007_348_a9,
     author = {A. P. Shcheglova},
     title = {The {Neumann} problem for semilinear elliptic equation in thin cylinder.},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {272--302},
     publisher = {mathdoc},
     volume = {348},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_348_a9/}
}
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A. P. Shcheglova. The Neumann problem for semilinear elliptic equation in thin cylinder.. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 38, Tome 348 (2007), pp. 272-302. http://geodesic.mathdoc.fr/item/ZNSL_2007_348_a9/