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On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds

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  • We investigate the uniqueness of nonnegative solutions to the following differential inequality \begin{eqnarray} div(A(x)|\nabla u|^{m-2}\nabla u)+V(x)u^{\sigma_1}|\nabla u|^{\sigma_2}\leq0, \tag{1} \end{eqnarray} on a noncompact complete Riemannian manifold, where $A, V$ are positive measurable functions, $m>1$, and $\sigma_1$, $\sigma_2\geq0$ are parameters such that $\sigma_1+\sigma_2>m-1$.
    Our purpose is to establish the uniqueness of nonnegative solution to (1) via very natural geometric assumption on volume growth.
    Mathematics Subject Classification: 35J70, 58J05.


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