分数阶拉普拉斯方程周期解的多重性

The Multiplicity of Periodic Solutions for Fractional Laplacian Equation

摘要: 本文研究以下分数阶拉普拉斯方程的周期解${\left(\begin{array}{c}-\Delta \end{array}\right)}^{s}u\left(x\right)+F^{\prime }\left(u\right)=0,u\left(x\right)=u\left(x+T\right),$ 其中$F:ℝ\to ℝ$ 是充分光滑的函数。当函数$F$ 满足恰当条件，我们利用变分方法与截断技巧，证明对于任意周期$T>0$ ，上述方程均有无穷多周期为$T$ 的周期解。

Abstract: We consider the periodic solutions of the following fractional Laplacian equation ${\left(\begin{array}{c}-\Delta \end{array}\right)}^{s}u\left(x\right)+F^{\prime }\left(u\right)=0,u\left(x\right)=u\left(x+T\right),$ where $F:ℝ\to ℝ$ is smooth enough function. Under suitable condition of $F$ , by employing variational method and truncation method, we prove for any periodic $T>0$ , the above equation has infinity periodic solutions with period $T$ .