标题： 奇异分数双相问题与变指数的莫尔斯理论 显示英文标题

标题： Singular fractional double-phase problems with variable exponent via Morse's theory

摘要： 在本文中，我们研究了一类涉及奇异项和消失电位的分数非局部问题，形式如下： \begin{eqnarray*} \begin{gathered} \left\{\begin{array}{llll} \mathcal{L}^{s_{1}, s_{2}}_{p(\mathrm{x}, .), q(\mathrm{x}, .)}\mathrm{w}(\mathrm{x})&= \displaystyle\frac{g(\mathrm{x}, \mathrm{w}(\mathrm{x}))}{ \mathrm{w}(\mathrm{x})^{\xi(\mathrm{x})}} + \mathcal{V}(\mathrm{x}) \vert \mathrm{w}(\mathrm{x}) \vert^{\sigma(\mathrm{x})-2} \mathrm{w}(\mathrm{x}) & \text { in } & \mathcal{U}, \\ \hspace{2cm} \mathrm{w}&> 0 & \text { in }& \mathcal{U},\\ \hspace{2cm} \mathrm{w}&=0 & \text { in }& \mathbb{R}^{N} \backslash \mathcal{U}, \end{array}\right. \end{gathered} \end{eqnarray*}其中，$ \mathcal{L}^{s_{1}, s_{2}}_{p(\mathrm{x}, .), q(\mathrm{x}, .)}$是一个 $\left(p(\mathrm{x}, .), q(\mathrm{x}, .)\right)$-分数双相算子，其中$ s_{1},s_{2 }\in \left( 0, 1\right)$，$g,$和$\mathcal{V}$是满足某些条件的函数。 证明这些结果的策略是通过近似方法处理问题并计算临界群。 此外，使用莫尔斯理论证明我们的问题有无限多解。 显示英文摘要

摘要： In this manuscript, we deal with a class of fractional non-local problems involving a singular term and vanishing potential of the form: \begin{eqnarray*} \begin{gathered} \left\{\begin{array}{llll} \mathcal{L}^{s_{1}, s_{2}}_{p(\mathrm{x}, .), q(\mathrm{x}, .)}\mathrm{w}(\mathrm{x})&= \displaystyle\frac{g(\mathrm{x}, \mathrm{w}(\mathrm{x}))}{ \mathrm{w}(\mathrm{x})^{\xi(\mathrm{x})}} + \mathcal{V}(\mathrm{x}) \vert \mathrm{w}(\mathrm{x}) \vert^{\sigma(\mathrm{x})-2} \mathrm{w}(\mathrm{x}) & \text { in } & \mathcal{U}, \\ \hspace{2cm} \mathrm{w}&> 0 & \text { in }& \mathcal{U},\\ \hspace{2cm} \mathrm{w}&=0 & \text { in }& \mathbb{R}^{N} \backslash \mathcal{U}, \end{array}\right. \end{gathered} \end{eqnarray*} where, $ \mathcal{L}^{s_{1}, s_{2}}_{p(\mathrm{x}, .), q(\mathrm{x}, .)}$ is a $\left(p(\mathrm{x}, .), q(\mathrm{x}, .)\right)$-fractional double-phase operator with $ s_{1},s_{2 }\in \left( 0, 1\right)$, $g,$ and $\mathcal{V}$ are functions that satisfy some conditions. The strategy of the proof for these results is to approach the problem proximatively and calculate the critical groups. Moreover, using Morse's theory to prove our problem has infinitely many solutions.