# American Institute of Mathematical Sciences

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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 and Continuous Dynamical Systems, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083
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