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Volume 10, Number 6, 2020, Pages 2704-2710                                                                DOI:10.11948/20200083
Solutions for the Kirchhoff type equations with fractional Laplacian
Yanping Jia,Ying Gao,Guang Zhang
Keywords:Kirchhoff equation, fractional Laplacian, equivalence,uniqueness.
Abstract:
      Due to the singularity and nonlocality of the fractional Laplacian, the classical tools such as Sturm comparison, Wronskians, Picard--Lindel\"{o}f iteration, and shooting arguments (which are all purely local concepts) are not{\ applicable} when analyzing solutions in the setting of the nonlocal operator $\left( -\Delta \right) ^{s}$. Furthermore, the nonlocal term of the Kirchhoff type equations will also cause some mathematical difficulties. The present work is motivated by the method of semi-classical problems which show that the existence of solutions of the Kirchhoff type equations are equivalent to the corresponding associated fractional differential and algebraic system. In such case, the existence of the fractional Kirchhoff equation can be obtained by using the corresponding fractional elliptic equation. Therefore some qualitative properties of solutions for the associated problems can be inherited. In particular, the classical uniqueness results can be applied to this equation.
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