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 ?
n , 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 ∈ ?. Finally, a regularity investigation in the plane shows that the singular points in
\(\mathcal {S}_{\kappa }\) are isolated.
