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

TL;DR

This paper proves the existence of solutions for a complex elliptic PDE involving a variable-order fractional Laplacian, singular and discontinuous nonlinearities, and a Choquard term, extending the theory to critical variable exponents.

Contribution

It introduces a novel existence result for a mixed operator elliptic problem with singular, discontinuous, and nonlocal nonlinearities involving variable exponents and critical growth.

Findings

Existence of solutions for the complex elliptic problem.

Convergence of solutions as the parameter approaches zero.

Extension of solution theory to variable order fractional operators.

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 $\alpha=0$.