On least Energy Solutions to A Semilinear Elliptic Equation in A Strip

Abstract

We consider the following semilinear elliptic equation on a strip: \[ \left\{{array}{l} Δu-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0, \frac{\partial u}{\partial ν}=0 \ {on} \ \partial (\R^{N-1} \times (0, L)) {array} \right.\] where $1<p\le \frac{N+2}{N-2}$. When $1<p<\frac{N+2}{N-2}$, it is shown that there exists a unique $L_{\ast }>0$ such that for $L\le L_{\ast }$, the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for $L>L_{\ast }$, the least energy solution is nontrivial. When $N\ge 4,p=\frac{N+2}{N-2}$, it is shown that there are two numbers $L_{\ast }<L_{\ast \ast }$ such that the least energy solution is trivial when $L\le L_{\ast }$, the least energy solution is nontrivial when $L\in (L_{\ast },L_{\ast \ast }]$, and the least energy solution does not exist when $L>L_{\ast \ast }$. A connection with Delaunay surfaces in CMC theory is also made.

typos corrected and uniqueness added