Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-Δ)^{s(\cdot)}u+b(-Δ)u&=λ|u|^{-γ(x)-1}u+\left(\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u) & +ηH(u-α)|u|^{r(x)-2}u,~$\text{in}$~Ω, u&=0,~$\text{in}$~$\mathbb{R}$^N\setminusΩ. \end{split} \end{align*} where $a(-\mathrm{\Delta }{)}^{s(\cdot )}+b(-\mathrm{\Delta })$ is a mixed operator with variable order $s(\cdot ):{\mathbb{R}}^{2N}\to (0,1)$, $a,b\ge 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\le 0$, $H(t)=1$ if $t>0),$ $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain, $N\ge 2$, $\lambda >0$, $0<{\gamma }^{-}=\{\gamma (x)\}\le \gamma (x)\le {\gamma }^{+}=\{\gamma (x)\}<1$, $\mu$ is a continuous variable parameter, and $F$ is the primitive function of a suitable $f$. The variable exponent $r(x)$ can be equal to the critical exponent $2_s^{\ast }(x)=\frac{2N}{N-2\bar{s}(x)}$ with $\bar{s}(x)=s(x,x)$ for some $x\in \bar{\mathrm{\Omega }},$ and $\eta$ is a positive parameter. We also show that as $\alpha \to 0^{+}$, the corresponding solution converges to a solution for the above problem with $\alpha =0$.