One-dimensional symmetry of positive bounded solutions to the nonlinear Schr\"{o}dinger equation in the half-space

TL;DR

This paper proves that for a specific nonlinear Schrödinger equation in a half-space, the only bounded positive solutions are one-dimensional when the boundary value equals a critical constant, extending symmetry results.

Contribution

It establishes the one-dimensional symmetry of bounded positive solutions at the critical boundary value for dimensions two and three.

Findings

Uniqueness of solutions at the critical boundary value in 2D and 3D.

No additional bounded positive solutions besides the one-dimensional solution at critical value.

Extension of symmetry results to the critical case in low dimensions.

Abstract

We are concerned with the half-space Dirichlet problem \[\begin{array}{ll} -\Delta v+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array} \] where $\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and $p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, such that for $0<c<c_p$ there are infinitely many bounded positive solutions, whereas, for $c>c_p$ there are no bounded positive solutions. If $N=2,\ 3$, we show that in the case $c = c_p$ there is no other bounded positive solution besides the one-dimensional one.