The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem "Equation missing"where $$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$ R + N : = { x ∈ R N : x N > 0 } for some $$N \ge 1$$ N ≥ 1 and $$p > 1$$ p > 1 , $$c > 0$$ c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ). We prove that the existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ) depend in a striking way on the value of $$c > 0$$ c > 0 and also on the dimension N. We find an explicit number $${c_p}\in (1,\sqrt{e})$$ c p ∈ ( 1 , e ) , depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions $$N \ge 2$$ N ≥ 2 , we prove that, for $$0< c < {c_p}$$ 0 < c < c p , problem ($$P_c$$ P c ) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$ c > c p , there are no bounded positive solutions to ($$P_c$$ P c ).