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Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory

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  • Uniqueness of nonzero positive solutions of a Laplacian elliptic equation arising in combustion theory is of great interest in combustion theory since it can be applied to determine where the extinction phenomenon occurs. We study the uniqueness whenever the orders of the reaction rates are in $(-\infty,1]$. Previous results on uniqueness treated the case when the orders belong to $[0,1)$. When the orders are negative or 1, it is physically meaningful and the bimolecular reaction rate corresponds to the order 1, but there is little study on uniqueness. Our results on the uniqueness are completely new when the orders are negative or 1, and also improve some known results when the orders belong to $(0,1)$. Our results provide exact intervals of the Frank-Kamenetskii parameters on which the extinction phenomenon never occurs. The novelty of our methodology is to combine and utilize the results from Laplacian elliptic inequalities and equations to derive new results on uniqueness of nonzero positive solutions for general Laplacian elliptic equations.
    Mathematics Subject Classification: Primary: 35J61; Secondary: 35R45, 49J40, 80A25.

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