One-dimensional symmetry of positive bounded solutions to the subcubic and cubic nonlinear Schrödinger equation in the half-space in dimensions $N=4,5$

Abstract

We are concerned with the half-space Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta v+v=|v|^{p-1}v &{} \text {in}\ {\mathbb {R}}^N_+, \\ v=c\ \text {on}\ \partial {\mathbb {R}}^N_+, &{}\lim _{x_N\rightarrow \infty }v(x',x_N)=0\ \text {uniformly in}\ x'\in {\mathbb {R}}^{N-1}, \end{array}\right. \end{aligned}$$

where ${\mathbb {R}}^N_+=\{x\in {\mathbb {R}}^N \ : \ x_N>0\}$ for some $N\ge 2$, 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 $2\le N\le 5$. Then, for the cubic NLS, where $p=3$, we establish this for $2\le N \le 4$. 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.

Appendix A. A Liouville type theorem

For the reader’s convenience, we state below a well known result due to [13] (see also [1] and [16,  Ch. I] for simpler proofs and extensions), which we used in the proofs of Theorems 1.1 and 1.2.

Theorem A.1

Let $1 < p \le p_{sg}(n)$, where

$$\begin{aligned} p_{sg}(n)=\left\{ \begin{array}{ll} \infty &{} \text {if}\ n\le 2, \\ &{} \\ \frac{n}{n-2} &{} \text {if}\ n> 2. \end{array} \right. \end{aligned}$$

Then, the inequality

$$\begin{aligned} -\Delta u \ge u^p,\ x\in {\mathbb {R}}^n, \end{aligned}$$

does not possess any positive classical solution.

Remark A.2

As was remarked in [16], the condition $p \le p_{sg}$ in Theorem A.1 is optimal, as shown by the explicit example $u(x) = k\left( 1 + |x|^2\right) ^{-1/(p-1)}$ with $n \ge 3$, $p > p_{sg}$ and $k > 0$ small enough.