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We study the nodal set of stationary solutions to equations of the form    $$(-\Delta )^s u = \lambda _+ (u_+)^{q-1} - \lambda _- (u_-)^{q-1}\quad \text {in }B_1,$$  ( - Δ ) s u = λ + ( u + ) q - 1 - λ - ( u - ) q - 1 in B 1 , where    $$\lambda _+,\lambda _-&gt;0, q \in [1,2)$$  λ + , λ - &gt; 0 , q ∈ [ 1 , 2 ) , and    $$u_+$$  u + and    $$u_-$$  u - are respectively the positive and negative part of  u . This collection of nonlinearities includes the unstable two-phase membrane problem    $$q=1$$  q = 1 as well as sublinear equations for    $$1&lt;q&lt;2$$  1 &lt; q &lt; 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    $$s=1$$  s = 1 , we prove that the admissible vanishing orders can not exceed the critical value    $$k_q= 2s/(2- q)$$  k q = 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    $$k_q&lt; 1$$  k q &lt; 1 , we prove a remarkable difference with the local case: solutions can only vanish with order    $$k_q$$  k q 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.

The nodal set of solutions to some nonlocal sublinear problems