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- Title
The nodal set of solutions to some nonlocal sublinear problems.
- Authors
Tortone, Giorgio
- Abstract
We study the nodal set of stationary solutions to equations of the form (- Δ) s u = λ + (u +) q - 1 - λ - (u -) q - 1 in B 1 , 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 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 , 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.
- Subjects
NODAL analysis; FINITE, The; CONTINUATION methods
- Publication
Calculus of Variations & Partial Differential Equations, 2022, Vol 61, Issue 3, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-022-02197-5