The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of stationary solutions to equations of the form (-Δ)su=λ+(u+)q-1-λ-(u-)q-1inB1, where λ+, λ-> 0 , q∈ [1 , 2) , and u+ and u- are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem q= 1 as well as sublinear equations for 1 < q< 2. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case s= 1 , we prove that the admissible vanishing orders can not exceed the critical value kq= 2 s/ (2 - q). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that kq< 1 , we prove a remarkable difference with the local case: solutions can only vanish with order kq and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.