The nodal set of solutions to some nonlocal sublinear problems

TL;DR

This paper investigates the structure and regularity of the nodal set of solutions to certain nonlocal fractional Laplacian equations with sublinear nonlinearities, establishing properties like unique continuation, vanishing order bounds, and stratification.

Contribution

It extends the analysis of nodal sets to nonlocal fractional problems with sublinear terms, including new regularity and stratification results, and identifies differences from the local case.

Findings

Proves strong unique continuation property.

Establishes finiteness of vanishing order and bounds it by a critical value.

Shows solutions can only vanish with a specific order and admits one-dimensional solutions.

Abstract

We study the nodal set of solutions to equations of the form $$ (-\Delta)^s u = \lambda_+ (u_+)^{q-1} - \lambda_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ 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.