We study the regularity of solutions of the following semilinear problem
$${\Delta}u = -\lambda_{+}(x) (u^{+})^{q}+\lambda_{-} (x) (u^{-})^{q} \qquad \text{in} \;\; B_{1}, $$
where B
1 is the unit ball in ℝn, 0 < q < 1 and λ
± satisfy a Hölder continuity condition. Our main results concern local regularity analysis of solutions and their nodal set {u = 0}. The desired regularity is C
[κ],κ−[κ] for κ = 2/(1 − q) and we divide the singular points in two classes. The first class contains the points where at least one of the derivatives of order less than κ is nonzero, the second class which is named \(\mathcal {S}_{\kappa }\), is the set of points where all the derivatives of order less than κ exist and vanish. We prove that \(\mathcal {S}_{\kappa }=\varnothing \) when κ is not an integer. Moreover, with an example we show that \(\mathcal {S}_{\kappa }\) can be nonempty if κ ∈ ℕ. Finally, a regularity investigation in the plane shows that the singular points in \(\mathcal {S}_{\kappa }\) are isolated.