The nodal set of solutions to some nonlocal sublinear problems

Abstract

We study the nodal set of solutions to equations of the form $$(-\Delta {)}^{s}u={\lambda }_{+}(u_{+}{)}^{q-1}-{\lambda }_{-}(u_{-}{)}^{q-1}\text{in B1},$$ where ${\lambda }_{+},{\lambda }_{-}>0,q\in [1,2)$, 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 $k_q=2s/(2-q)$. Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that $k_q<1$, we prove a remarkable difference with the local case: solutions can only vanish with order $k_q$ 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.

53 pages