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

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.