The Singular Set for a Semilinear Unstable Problem

Abstract

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.