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

Zuo, Jiabin (author), Choudhuri, Debajyoti (author), Repovš, Dušan (author)

We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent $\begin{aligned} \begin{aligned} a(-\varDelta )^{s(\cdot )}u+b(-\varDelta )u&=\lambda |u|^{-\gamma (x)-1}u+\left( \int _{\varOmega }\frac{F(y,u(y))}{|x-y| ^{\mu (x,y)}}dy\right) f(x,u)\\&+\eta H(u-\alpha )|u|^{r(x)-2}u,~\text {in}~\varOmega ,\\ u&=0,~\text {in}~{\mathbb {R}}^N\setminus \varOmega, \end{aligned} \end{aligned}$ where $a(-\varDelta )^{s(\cdot )}+b(-\varDelta )$ is a mixed operator with variable order $s(\cdot ):{\mathbb {R}}^{2N}\rightarrow (0,1)$, $a, b\ge 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\le 0$, $H(t)=1$ if $t>0$), $\varOmega \subset {\mathbb {R}}^N$ is a bounded domain, $N\ge 2$, $\lambda >0$, $0<\gamma ^{-}=\underset{x\in \bar{\varOmega }}{\inf }\{\gamma (x)\}\le \gamma (x)\le \gamma ^+ =\underset{x\in \bar{\varOmega }}{\sup }\{\gamma (x)\}<1$, $\mu$ 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 $x \in \bar{\varOmega }$, 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 $\alpha =0$.

• Language: English
• Keywords: Choquard type, variable-order fractional operator, mixed operator, variable singular exponent, discontinuous power nonlinearity
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Author Accepted Manuscript
• Year: 2022
• Number of pages: Str. 2532-2553
• Numbering: Vol. 25, iss. 6
• PID: 20.500.12556/RUL-143428
• UDC: 517.956
• ISSN on article: 1311-0454
• DOI: 10.1007/s13540-022-00105-4
• COBISS.SI-ID: 130585603
• Publication date in RUL: 20.12.2022

Record is a part of a journal

• Title: Fractional Calculus & Applied Analysis
• Shortened title: Fract. Calc. Appl. Anal.
• Publisher: De Gruyter, Bulgarian Academy of Sciences. Institute of Mathematics and Informatics
• ISSN: 1311-0454
• COBISS.SI-ID: 18093913