  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.
