Radial limits of positive solutions to the Darboux equation

Assume that a positive function $u$ satisfies the Darboux equation
$$ \Delta u=\frac{(\alpha-1)}y\frac{\partial u}{\partial y},\qquad\alpha>0, $$
in the upper half-space $\mathbb R_+^{d+1}$. We investigate Bloch type conditions that guarantee the following property: for any $a\in(0,+\infty)$, the set where the radial limit of $u$ is equal to $a$, is large in the sense of the Hausdorff dimension. Bibl. – 6 titles.