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

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

2010-10-12

arxiv.org/abs/1010.2289v1

DCDS28(2010),no.3,1083-1099

Mathematics

Analysis of PDEs

typos corrected and uniqueness added

Scientific paper

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

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 $L_{*}L_{**}$. A connection with Delaunay surfaces in CMC theory is also made.

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