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Solutions for the Kirchhoff type equations with fractional Laplacian
Guang Zhang
Keywords:Kirchhoff equation, Fractional Laplacian, equivalence, uniqueness.
      Due to the singularity and nonlocality of the fractional Laplacian, the classical tools such as Sturm comparison, Wronskians, Picard--Lindel?f iteration, and shooting arguments (which are all purely local concepts) are not at our disposal when analyzing solutions in the setting of the nonlocal operator (-Δ)^{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 and some recent works to obtain that solutions of the Kirchhoff type equations is equivalent to the corresponding associated differential and algebraic system. In this case, the nonlocal term of the Kirchhoff operator can be removed. So then, some qualitative properties of solutions for the associated problems can be inherited. In particular, the classical uniqueness result can be expanded.