The nonlinear Schrödinger equation in the half-space

Abstract

The present paper is concerned with the half-space Dirichlet problem \begin{equation} \tag{$P_c$} \label{problem-abstract} -Δv + v = |v|^{p-1}v,\ \mbox{ in } \mathbb{R}^N_{+}, \qquad v = c,\ \mbox{ on } \partial \mathbb{R}^N_{+},\ \qquad \lim_{x_N \to \infty} v(x',x_N) = 0 \mbox{ uniformly in }x' \in \mathbb{R}^{N-1}, \end{equation} where ${\mathbb{R}}_{+}^N:=\{x\in {\mathbb{R}}^N:x_N>0\}$ for some $N\ge 1$ and $p>1$, $c>0$ are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to \eqref{problem-abstract}. We prove that the existence and multiplicity of bounded positive solutions to \eqref{problem-abstract} depend in a striking way on the value of $c>0$ and also on the dimension $N$. We find an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, which determines the threshold between existence and non-existence. In particular, in dimensions $N\ge 2$, we prove that, for $0<c<c_p$, problem \eqref{problem-abstract} admits infinitely many bounded positive solutions, whereas, for $c>c_p$, there are no bounded positive solutions to \eqref{problem-abstract}.

Minor changes have been made; to appear in "Mathematische Annalen"