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

Abstract

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(-\Delta)^{s(\cdot)}u+b(-\Delta)u&=\lambda |u|^{-\gamma(x)-1}u+\left(\int_{\Omega}\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}~\Omega, u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega. \end{split} \end{align*} where $a(-\Delta)^{s(\cdot)}+b(-\Delta)$ is a mixed operator with variable order $s(\cdot):\mathbb{R}^{2N}\rightarrow (0,1)$, $a, b\geq 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\leq0$, $H(t) = 1$ if $t>0),$ $\Omega\subset\mathbb{R}^N$ is a bounded domain, $N\geq 2$, $\lambda>0$, $0<\gamma^{-}=\underset{x\in\bar{\Omega}}{\inf}\{\gamma(x)\}\leq\gamma(x)\leq\gamma^+=\underset{x\in\bar{\Omega}}{\sup}\{\gamma(x)\}<1$, $\mu$ 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 $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$ with $\bar{s}(x)=s(x,x)$ for some $x\in\bar{\Omega},$ and $\eta$ is a positive parameter. We also show that as $\alpha\rightarrow 0^+$, the corresponding solution converges to a solution for the above problem with $\alpha=0$.