The free boundary for a semilinear non-homogeneous Bernoulli problem

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a "regular" part and a "singular" part, as Alt and Caffarelli have shown in their pioneer work (J. Reine Angew. Math., 325, 105-144, 1981) that regular points are $C^{1,\gamma}$ in two-dimensions. Later, Weiss (J. Geom. Anal., 9, 317-326, 1999) first realized that in higher dimensions a critical dimension $d^{*}$ exists so that the singularities of the free boundary can only occur when $d\geqslant d^{*}$. In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a $C^{1,\gamma}$ function for some $\gamma\in(0,1)$. In addition, there exists a critical dimension $d^{*}$ so that the singular set is empty if $d<d^{*}$, discrete if $d=d^{*}$ and of locally finite $\mathcal{H}^{d-d^{*}}$ Hausdorff measure if $d>d^{*}$. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.