  We consider the following semilinear elliptic equation on a strip: \[
\left\{{array}{l} \Delta u-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0,
\frac{\partial u}{\partial \nu}=0 \ {on} \ \partial (\R^{N-1} \times (0, L))
{array} \right.\] 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.
