具有双临界指数的分数阶Kirchhoff方程的正规化解的非存在性结果

A Nonexistence Result of the Normalized Solutions to a Fractional Kirchhoff Equation with Doubly Critical Exponents

Kirchhoff模型源于研究一根有弹性的绳子在自由振动过程中绳长的改变量,而分数阶Kirchhoff方程则将局部问题延伸到了非局部问题。通过使用变分法,约束极小元思想和一些能量估计,本文证明了一类具有双临界指数和混合非线性项的分数阶Kirchhoff方程的正规化解的非存在性结果,即泛函在一个L2-约束流形上的能量极小元的不存在性。这些能量估计是本文的重点和难点内容,但本文仅针对非存在性结果进行了分析。对分数阶和非局部算子的研究不仅可以应用于数学领域,还能用于连续介质力学,相变现象,博弈论等其他方面。

The Kirchhoff model is derived from the study of the length change of an elastic rope during vibration, fractional Kirchhoff equations, however, extend local problems to nonlocal ones. By using the variational method, constrained minimization technique and some energy estimates, a nonexistence result of the normalized solutions to the fractional Kirchhoff equation with doubly critical exponents and combined nonlinearities is obtained in this paper. In other words, the nonexistence of energy minimizers of the functional on the L2-constrained monifold is discussed. These energy estimates are the key and difficult points of this paper. Nevertheless, only the nonexistence result is analyzed here. Besides, the study of fractional order and nonlocal operators can be applied not only to the field of Mathematics, but also to continuum mechanics, phase transition phenomena, game theory, etc.