One-dimensional symmetry of positive bounded solutions to the subcubic and cubic nonlinear Schr\"{o}dinger equation in the half-space in dimensions $N=4,5$

TL;DR

This paper proves the one-dimensional symmetry of positive bounded solutions to certain nonlinear Schrödinger equations in half-spaces for specific dimensions and nonlinearities, using a novel approach based on super-solution techniques.

Contribution

It establishes the uniqueness of one-dimensional solutions for the problem in dimensions 2 to 5 for subcubic and cubic nonlinearities, employing a new method involving auxiliary super-solutions.

Findings

Proved one-dimensional symmetry for 1<p<3 in 2≤N≤5.

Established symmetry for cubic NLS in 2≤N≤4.

Developed a new approach using super-solution and Liouville theorems.

Abstract

We are concerned with the half-space Dirichlet problem \[\left\{\begin{array}{ll} -\Delta v+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array}\right. \] 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. They also posed as an interesting open question whether the one-dimensional solution is the unique bounded positive solution in the case where $c = c_p$. If $N=2, 3$, we recently showed this one-dimensional symmetry property in [Partial Differ. Equ. Appl. (2021)] by adapting some ideas from the proof of De Giorgi's conjecture in low dimensions. Here, we first focus on the case $1<p<3$ and prove this uniqueness property in dimensions $2\leq N\leq 5$. Then, for the cubic NLS, where $p = 3$, we establish this for $2 \leq N \leq 4$. Our approach is completely different and relies on showing that a suitable auxiliary function, inspired by a Lyapunov-Schmidt type decomposition of the solution, is a nonnegative super-solution to a Lane-Emden-Fowler equation in $\mathbb{R}^{N-1}$, for which an optimal Liouville type result is available.