We consider the Laplace equation in the half-space satisfying a nonlinear Neumann condition with boundary potential. This class of problems appears in a number of mathematical and physics contexts and is linked to fractional dissipation problems. Here the boundary potential and nonlinearity are singular and of power-type, respectively. Depending on the degree of singularity of potentials, first we show a nonexistence result of positive solutions in
D
1
,
2
(
R
+
n
)
\mathcal {D}^{1,2}(\mathbb {R}^n_+)
with a
L
p
L^p
-type integrability condition on
∂
R
+
n
\partial \mathbb {R}^n_{+}
. After, considering critical nonlinearities and conditions on the size and sign of potentials, we obtain the existence of positive solutions by means of minimization techniques and perturbation methods.