In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations ${\left(a+b{\int }_{\Omega \times \Omega }\frac{{|\xi \left(x\right)-\xi \left(y\right)|}^{{}^p}}{{|x-y|}^{N+ps(x,y)}}dxdy\right)}^{p-1}{(-\Delta )}_p^{s(·)}\xi +\lambda V\left(x\right){\left|\xi \right|}^{p-2}\xi =f(x,\xi ),x\in \Omega ,\xi =0,x\in \partial \Omega ,$ where $\Omega$ is a bounded Lipschitz domain in ${\mathbb{R}}^N$, $1<p<+\infty$, $a,b>0$ are constants, $s(·):{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,1)$ is a continuous and symmetric function with $N>s(x,y)p$ for all $(x,y)\in \Omega \times \Omega$, $\lambda >0$ is a parameter, ${(-\Delta )}_p^{s(·)}$ is a fractional p-Laplace operator with variable-order, $V\left(x\right):\Omega \to {\mathbb{R}}^{+}$ is a potential function, and $f(x,\xi ):\Omega \times {\mathbb{R}}^N\to \mathbb{R}$ is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

中文翻译：

---

具有对称变阶的分数 p-Laplacian Schrödinger-Kirchhoff 型方程的无穷多解

在本文中，我们首先获得了变阶 Sobolev 空间的嵌入结果，然后我们考虑以下 Schrödinger-Kirchhoff 型方程 ${\left(\mathrm{一种}+乙{\int }_{\Omega \times \Omega }\frac{{|\xi \left(X\right)-\xi \left(是\right)|}^{{}^{磷}}}{{|X-是|}^{N+磷秒(X,是)}}dXd是\right)}^{磷-1}{(-\Delta )}_{磷}^{秒(·)}\xi +\lambda 伏\left(X\right){\left|\xi \right|}^{磷-2}\xi =F(X,\xi ),X\in \Omega ,\xi =0,X\in \partial \Omega ,$ 在哪里 $\Omega$ 是一个有界的 Lipschitz 域 ${\mathbb{电阻}}^N$, $1<磷<+\infty$, $\mathrm{一种},乙>0$ 是常数， $秒(·)：{\mathbb{电阻}}^N\times {\mathbb{电阻}}^N\to (0,1)$ 是一个连续且对称的函数 $N>秒(X,是)磷$ 对全部 $(X,是)\in \Omega \times \Omega$, $\lambda >0$ 是一个参数， ${(-\Delta )}_{磷}^{秒(·)}$是一个变阶的小数p -Laplace 算子，$伏\left(X\right)：\Omega \to {\mathbb{电阻}}^{+}$ 是一个势函数，并且 $F(X,\xi )：\Omega \times {\mathbb{电阻}}^N\to \mathbb{电阻}$是一个连续的非线性函数。假设V和f满足一些合理的假设，我们利用喷泉定理和对称山口定理，不使用 Ambrosetti-Rabinowitz（简称（AR））条件，得到上述问题的无穷多解的存在性。