A uniqueness theorem for harmonic functions on the upper-half plane

Consider harmonic functions on the upper-half plane $R^{2}_{+} = \{(x,y)\vert\;y > 0 \}$ satisfying the boundary condition $u_{y}=-\exp(u)$ and the constraint $\int_{R^{2}_{+}}\exp(2u) < \infty$. We prove that all such functions are of form (1.2) below.