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.