  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 $\mathbb{R}^N_{+} := \{\,x
\in \mathbb{R}^N: x_N > 0\, \}$ for some $N \geq 1$ 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 $N \geq 2$, 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}.
