  This paper concerns positive solutions to the boundary value problems of the
scalar field equation in the half space with a Sobolev supercritical
nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a
nontrivial nonnegative Radon measure as the boundary data. Under a suitable
integrability assumption on the boundary data and the Joseph--Lundgren
subcritical condition on the nonlinear term, we give a complete classification
of the existence/nonexistence of a positive solution with respect to the size
of the boundary data. Furthermore, we give a result on the existence of
multiple positive solutions via bifurcation theory.
