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

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

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 ▫$$\begin{aligned} \begin{aligned} a(-\varDelta )^{s(\cdot )}u+b(-\varDelta )u&=\lambda |u|^{-\gamma (x)-1}u+\left( \int _{\varOmega }\frac{F(y,u(y))}{|x-y| ^{\mu (x,y)}}dy\right) f(x,u)\\&+\eta H(u-\alpha )|u|^{r(x)-2}u,~\text {in}~\varOmega ,\\ u&=0,~\text {in}~{\mathbb {R}}^N\setminus \varOmega , \end{aligned} \end{aligned}$$▫ 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 ▫$\alpha =0$▫.

Keywords

Choquard type;variable-order fractional operator;mixed operator;variable singular exponent;discontinuous power nonlinearity;

Data

Language:	English
Year of publishing:	2022
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.956
COBISS:	130585603
ISSN:	1311-0454
Views:	134
Downloads:	26
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Embargo end date (OpenAIRE):	2023-12-01
Pages:	str. 2532-2553
Volume:	ǂVol. ǂ25
Issue:	ǂiss. ǂ6
Chronology:	Dec. 2022
DOI:	10.1007/s13540-022-00105-4
ID:	17469812