On least energy solutions to a semilinear elliptic equation in a strip

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$
$ u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L)) $

where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L _* >0 such that for L $\leq $L _* , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L _* , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L _* < L _** such that the least energy solution is trivial when L $\leq$L _*, the least energy solution is nontrivial when L $\in$(L _*,L _**], and the least energy solution does not exist when L >L _**. A connection with Delaunay surfaces in CMC theory is also made.