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 following critical Choquard type problem with a variable-order fractional Laplacian and variable singular exponent $$\begin{aligned} \begin{aligned} a(-\varDelta )^{s(\cdot )}u+b(-\varDelta )u&=\lambda |u|^{-\gamma (x)-1}u+\left( \int _{\varOmega }\frac{F(y,u(y))}{|x-y| ^{\mu (x,y)}}dy\right) f(x,u)\\&+\eta H(u-\alpha )|u|^{r(x)-2}u,~\text {in}~\varOmega ,\\ u&=0,~\text {in}~{\mathbb {R}}^N\setminus \varOmega . \end{aligned} \end{aligned}$$ where $$a(-\varDelta )}+b(-\varDelta )$$ is mixed operator order $$s(\cdot ):{\mathbb {R}}^{2N}\rightarrow (0,1)$$ , $$a, b\ge 0$$ $$a+b>0$$ H Heaviside function (i.e., $$H(t)=0$$ if $$t\le $$H(t) = 1$$ $$t>0),$$ $$\varOmega \subset {\mathbb {R}}^N$$ bounded domain, $$N\ge 2$$ $$\lambda >0$$ $$0<\gamma ^{-}=\underset{x\in \bar{\varOmega }}{\inf }\{\gamma (x)\}\le \gamma (x)\le ^+ =\underset{x\in }}{\sup (x)\}<1$$ $$\mu $$ continuous parameter, F primitive suitable f. The r(x) can be equal to $$2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$$ $$\bar{s}(x)=s(x,x)$$ some $$x\in },$$ $$\eta positive parameter. also show that as $$\alpha \rightarrow 0^+$$ corresponding solution converges above =0$$