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

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  • 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.

    Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J40.

    Citation:

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