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

Abstract

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.