The nonlinear Schrödinger equation in the half-space

Abstract 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 ∈ > 0 } for some $$N \ge 1$$ ≥ 1 and $$p p , $$c 0$$ c are constants. We analyse existence, non-existence multiplicity of bounded positive solutions to ( $$P_c$$ P ). prove that existence ) depend in a striking way on value also dimension N . find an explicit number $${c_p}\in (1,\sqrt{e})$$ ( , e ) depending only p which determines threshold between non-existence. In particular, dimensions 2$$ 2 we that, $$0< c < {c_p}$$ < admits infinitely many solutions, whereas, there