The nonlinear Schr\"odinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem \begin{equation} \tag{$P_c$} \label{problem-abstract} -\Delta v + 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, }$forsome$N \geq 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}.