The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem 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 ( $$P_c$$ ). We prove that the existence and multiplicity of bounded positive solutions to ( $$P_c$$ ) 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 ( $$P_c$$ ) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$ , there are no bounded positive solutions to ( $$P_c$$ ).