The nodal set of solutions to some nonlocal sublinear problems

Abstract

AbstractWe 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 $$\lambda _+,\lambda _->0, q \in [1,2)$$ λ + , λ - > 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 $$1<q<2$$ 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 $$s=1$$ 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 $$k_q< 1$$ k q < 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.