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Superlinear elliptic inequalities on manifolds
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2020-05-01 , DOI: 10.1016/j.jfa.2019.108444
Alexander Grigor'yan , Yuhua Sun , Igor Verbitsky

Let $M$ be a complete non-compact Riemannian manifold and let $\sigma $ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \begin{equation*} -\Delta u\geq \sigma u^{q}\quad \text{in}\,\,M, \end{equation*} where $q>1$. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of $\Delta $. In particular, explicit necessary and sufficient conditions are given when $M$ has nonnegative Ricci curvature everywhere in $M$, or more generally when Green's function satisfies the 3G-inequality.

中文翻译:

流形上的超线性椭圆不等式

令 $M$ 是一个完整的非紧黎曼流形,并令 $\sigma $ 是对 $M$ 的氡测度。我们研究半线性椭圆不等式的正解存在或不存在的问题 \begin{equation*} -\Delta u\geq \sigma u^{q}\quad \text{in}\,\,M, \end{equation*} 其中 $q>1$。我们根据 $\Delta $ 的格林函数获得了存在正解的必要和充分标准。特别地,当 $M$ 在 $M$ 中处处具有非负 Ricci 曲率时,或者更一般地说,当格林函数满足 3G 不等式时,给出了明确的充分必要条件。
更新日期:2020-05-01
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