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July  2010, 28(3): 1083-1099. doi: 10.3934/dcds.2010.28.1083

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

1. 

Ecole des hautes etudes en sciences sociales, CAMS, 54, boulevard Raspail, F - 75006 - Paris, France

2. 

Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  March 2010 Revised  April 2010 Published  April 2010

We consider the following semilinear elliptic equation on 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.

Citation: Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083
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