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Non-existence of dead cores in fully nonlinear elliptic models
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2021-05-10 , DOI: 10.1142/s0219199721500395
João Vitor da Silva 1 , Disson dos Prazeres 2 , Humberto Ramos Quoirin 3
Affiliation  

We investigate non-existence of non-negative dead core solutions for the problem |Du|γF(x,D2u)+a(x)uq=0in Ω,u=0on Ω. Here, ΩN is a bounded smooth domain, F is a fully nonlinear elliptic operator, a:Ω is a sign-changing weight, γ0, and 0<q<γ+1. We show that this problem has no nontrivial dead core solutions if either q is close enough to γ+1 or the negative part of a is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on q and a. Our results extend previous ones established in the semilinear case, and are new even for the simple model |Du(x)|γTr(A(x)D2u(x))+a(x)uq(x)=0, where AC0(Ω;Sym(N)) is a uniformly elliptic and non-negative matrix.



中文翻译:

完全非线性椭圆模型中不存在死核

我们调查了该问题不存在非负死核解决方案|D|γF(X,D2)+一个(X)q=0在 Ω,=0上 Ω.这里,Ωñ是一个有界平滑域,F是一个完全非线性的椭圆算子,一个Ω是一个符号变化的权重,γ0, 和0<q<γ+1. 我们证明这个问题没有非平凡的死核解决方案,如果要么q足够接近γ+1或负面的部分一个足够小。此外,我们在这些条件下获得了正解的存在性和唯一性q一个. 我们的结果扩展了之前在半线性情况下建立的结果,即使对于简单模型也是新的|D(X)|γTr(一个(X)D2(X))+一个(X)q(X)=0, 在哪里一个C0(Ω;符号(ñ))是均匀椭圆非负矩阵。

更新日期:2021-05-10
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