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

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

Povzetek

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 ▫$\alpha =0$▫.

Ključne besede

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

Podatki

Jezik:	Angleški jezik
Leto izida:	2022
Tipologija:	1.01 - Izvirni znanstveni članek
Organizacija:	UL FMF - Fakulteta za matematiko in fiziko
UDK:	517.956
COBISS:	130585603
ISSN:	1311-0454
Št. ogledov:	134
Št. prenosov:	26
Ocena:	0 (0 glasov)
Metapodatki:

Ostali podatki

Vrsta dela (COBISS):	Članek v reviji
Konec prepovedi (OpenAIRE):	2023-12-01
Strani:	str. 2532-2553
Letnik:	ǂVol. ǂ25
Zvezek:	ǂiss. ǂ6
Čas izdaje:	Dec. 2022
DOI:	10.1007/s13540-022-00105-4
ID:	17469812