The Singular Set for a Semilinear Unstable Problem

We study the regularity of solutions of the following semilinear problemΔu=−λ+(x)(u+)q+λ−(x)(u−)qinB1,where B₁ 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 S_κ, is the set of points where all the derivatives of order less than κ exist and vanish. We prove that S_κ= ∅ when κ is not an integer. Moreover, with an example we show that S_κ can be nonempty if κ ∈ ℕ. Finally, a regularity investigation in the plane shows that the singular points in S_κ are isolated.