Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1083 - 1099, Volume 28, Issue 3, November 2010      doi:10.3934/dcds.2010.28.1083

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Henri Berestycki - Ecole des hautes etudes en sciences sociales, CAMS, 54, boulevard Raspail, F - 75006 - Paris, France (email)
Juncheng Wei - Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China (email)

Abstract: 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.

Keywords:  Semilinear Elliptic Equations, Unbounded Domains, Strip, Least Energy Solutions, Critical Sobolev Exponent.
Mathematics Subject Classification:  Primary: 35B40, 35B45; Secondary: 35J40.

Received: March 2010;      Revised: April 2010;      Available Online: April 2010.