Infinitely Many Solutions for Singular p(x)-Laplacian Equation Without the Ambrosetti–Rabinowitz (AR) Condition

We establish the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation with singularity:

$\begin{array}{cc}{(-\Delta )}_{p(·)}^{s}u& =\frac{\lambda }{|u{|}^{\gamma \left(x\right)-1}u}+f(x,u)\text{in}\Omega ,\\ u& =0\text{in}{ℝ}^N\setminus \Omega ,\end{array}$

where $\Omega \subset {ℝ}^N$ is a smooth bounded domain with $N\ge 2$. The parameters in the equation are as follows: $\lambda >0$ is a positive constant, $s\in (0,1)$ is a fixed parameter, $\gamma :\bar{\Omega }\to (0,1)$ is a continuous function, $N>sp(x,y)$ for all $(x,y)\in \bar{\Omega }\times \bar{\Omega }$, and ${(-\Delta )}_{p(·)}^s$ is the fractional $p(·)$-Laplacian operator with a variable exponent. The nonlinearity $f(x,u)$ is a Carathéodory function satisfying $f\ge 0$ and certain growth conditions. Furthermore, we establish a uniform $L^{\infty }\left(\bar{\Omega }\right)$ estimate for the solution(s) using the Moser iteration technique.