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

TL;DR

This paper investigates the existence and uniqueness of least energy solutions to a semilinear elliptic equation in a strip, revealing critical lengths where solutions change from trivial to nontrivial, and connecting these findings to geometric surfaces.

Contribution

It provides a detailed analysis of how the strip length influences the nature of least energy solutions, including existence, uniqueness, and nonexistence, with novel connections to Delaunay surfaces in CMC theory.

Findings

Existence of a critical length L* for subcritical p where solutions switch from trivial to nontrivial.

Identification of two critical lengths L* and L** for the critical exponent case, dictating solution behavior.

Connection established between least energy solutions and Delaunay surfaces in constant mean curvature theory.

Abstract

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.