2020 Volume 10 Issue 6
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Yanping Jia, Ying Gao, Guang Zhang. SOLUTIONS FOR THE KIRCHHOFF TYPE EQUATIONS WITH FRACTIONAL LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2704-2710. doi: 10.11948/20200083
Citation: Yanping Jia, Ying Gao, Guang Zhang. SOLUTIONS FOR THE KIRCHHOFF TYPE EQUATIONS WITH FRACTIONAL LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2704-2710. doi: 10.11948/20200083

SOLUTIONS FOR THE KIRCHHOFF TYPE EQUATIONS WITH FRACTIONAL LAPLACIAN

  • Corresponding author: Email address:lxyzhg@tjcu.edu.cn(G. Zhang)
  • Fund Project: The authors were supported by National Natural Science Foundation of China (Nos. 11371277, 11871314) and National Science Foundation of Tianjin (No. 19JCYBJC30700)
  • Due to the singularity and nonlocality of the fractional Laplacian, the classical tools such as Sturm comparison, Wronskians, Picard--Lindelöf 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.
    MSC: 35R11, 35A15, 35J60, 47G20
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