论文标题
某些非局部sublerear问题的节点解决方案集
The nodal set of solutions to some nonlocal sublinear problems
论文作者
论文摘要
我们研究了$$（-Δ）^s u =λ_+（u _+）^{q-1} {q-1} - λ_-（u _--）^{q-1} \ quad \ text {in $ b_1 $}，$，$λ_+，λ_-- $，λ_----- $，λ_----- $ u _- $分别是$ u $的正和负部分。这一非线性集合包括不稳定的两阶段膜问题$ q = 1 $以及$ 1 <q <2 $的sublinear方程。我们最初证明了强大的独特延续属性和消失顺序的有限性的有效性，以实施对节点集的爆炸分析。与本地情况一样，$ s = 1 $，我们证明可接受的消失订单不能超过关键值$ k_q = 2s/（2- q）$。此外，我们研究了节点集的规律性，并证明了分层结果。最终，对于$ k_q <1 $的这些参数，我们证明与本地案例有着显着的区别：解决方案只能通过订单$ k_q $消失，而问题则承认了一维解决方案。我们的方法是基于根据解决方案的消失顺序，基于Almgren型家族或魏斯型单调性公式的2-参数家族的有效性。
We study the nodal set of solutions to equations of the form $$ (-Δ)^s u = λ_+ (u_+)^{q-1} - λ_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $λ_+,λ_->0, q \in [1,2)$, 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.
