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  • Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents
    Zuo, Jiabin ; Choudhuri, Debajyoti ; Repovš, Dušan, 1954-
    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 ▫$\alpha =0$▫.
    Vir: Fractional Calculus & Applied Analysis. - ISSN 1311-0454 (Vol. 25, iss. 6, Dec. 2022, str. 2532-2553)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2022
    Jezik - angleški
    COBISS.SI-ID - 130585603

    Povezava(-e):

    https://link.springer.com/article/10.1007/s13540-022-00105-4
    Repozitorij Univerze v Ljubljani – RUL
    DOI

vir: Fractional Calculus & Applied Analysis. - ISSN 1311-0454 (Vol. 25, iss. 6, Dec. 2022, str. 2532-2553)
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