In this paper, we study the existence of multiple periodic solutions for the following fractional equation: (-Δ)s⁢u+F′⁢(u)=0,u⁢(x)=u⁢(x+T) x∈ℝ.(-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}. For an even double-well potential, we establish more and more periodic solutions for a large period T . Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

中文翻译：

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一类分数阶拉普拉斯方程组的多重周期解

本文研究以下分数阶方程的多重周期解的存在：（-Δ）s⁢u+ F′⁢（u）= 0，u⁢（x）=u⁢（x + T）x∈ ℝ。（-\ Delta）^ {s} u + F ^ {\ prime}（u）= 0，\ qquad u（x）= u（x + T）\ quad x \ in \ mathbb {R}。为了获得甚至双井潜力，我们针对大周期T建立越来越多的周期解。没有F的均匀性，我们给出了问题的两个周期解的存在。我们利用变分论证，特别是克拉克定理和莫尔斯理论。