The nodal set of solutions to some nonlocal sublinear problems
نویسندگان
چکیده
Abstract We study the nodal set of stationary solutions to equations form $$(-\Delta )^s u = \lambda _+ (u_+)^{q-1} - _- (u_-)^{q-1}\quad \text {in }B_1,$$ ( - Δ ) s u = λ + q 1 in B , where $$\lambda _+,\lambda _->0, q \in [1,2)$$ > 0 ∈ [ 2 , and $$u_+$$ $$u_-$$ are respectively positive negative part . This collection nonlinearities includes unstable two-phase membrane problem $$q=1$$ as well sublinear for $$1<q<2$$ < initially prove validity strong unique continuation property finiteness vanishing order, in order implement a blow-up analysis set. As local case $$s=1$$ we that admissible orders can not exceed critical value $$k_q= 2s/(2- q)$$ k / Moreover, regularity stratification result. Ultimately, those parameters such $$k_q< 1$$ remarkable difference with case: only vanish $$k_q$$ admits one dimensional solutions. Our approach is based on either family Almgren-type or 2-parameter Weiss-type monotonicity formulas, according solution.
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ژورنال
عنوان ژورنال: Calculus of Variations and Partial Differential Equations
سال: 2022
ISSN: ['0944-2669', '1432-0835']
DOI: https://doi.org/10.1007/s00526-022-02197-5