We are interested in the existence of solutions for the following fractional $p(x,\cdot )$ -Kirchhoff-type problem: $$ \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases} $$ where $\Omega \subset \mathbb{R}^{N}$ , $N\geq 2$ is a bounded smooth domain, $s\in (0,1)$ , $p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )$ , $(-\Delta )^{s}_{p(x,\cdot )}$ denotes the $p(x,\cdot )$ -fractional Laplace operator, $M: [0,\infty ) \to [0, \infty )$ , and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo–Benci–Fortunato (Nonlinear Anal. 7(9):981–1012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti–Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

中文翻译：

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没有Ambrosetti–Rabinowitz条件的一类分数\（p（x（\ cdot）\）- Kirchhoff型问题的多重解

我们对以下分数$ p（x，\ cdot）$ -Kirchhoff型问题的解决方案的存在感兴趣：$$ \ textstyle \ begin {cases} M（\ int _ {\ Omega \ times \ Omega} { \ frac {\ vert u（x）-u（y）\ vert ^ {p（x，y）}} {p（x，y）\ vert xy \ vert ^ {N + p（x，y）s} }} \，dx \，dy）（-\ Delta）^ {s} _ {p（x，\ cdot）} u = f（x，u），\ quad x \ in \ Omega，\\ u = 0 ，\ quad x \ in \ partial \ Omega，\ end {cases} $$，其中$ \ Omega \ subset \ mathbb {R} ^ {N} $，$ N \ geq 2 $是有界光滑域$ s \ in（0,1）$，$ p：\ overline {\ Omega} \ times \ overline {\ Omega} \ rightarrow（1，\ infty）$，$（-\ Delta）^ {s} _ {p（x ，\ cdot）} $表示$ p（x，\ cdot）$-分式Laplace运算符，$ M：[0，\ infty）\ to [0，\ infty）$和$ f：\ Omega \ times \ mathbb {R} \ to \ mathbb {R} $是连续函数。使用变分方法 尤其是由于Bartolo-Benci-Fortunato（非线性分析，7（9）：981-1012，1983年）所致的对称山口定理，我们在不假设Ambrosetti-Rabinowitz条件的情况下，建立了这个问题的无限多个解。我们在几个方向上的主要结果扩展了最近在文献中出现的先前结果。