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arXiv · 1010.2289

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

Henri Berestycki·Juncheng Wei
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Abstract

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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Henri Berestycki, Juncheng Wei. 2010-10-12. On least Energy Solutions to A Semilinear Elliptic Equation in A Strip. https://arxiv.org/abs/1010.2289

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