20.500.12556/RUL-143428 Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents 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 $\alpha =0$. Choquard type variable-order fractional operator mixed operator variable singular exponent discontinuous power nonlinearity true false true Angleški jezik Ni določen Članek v reviji 2022-12-20 13:58:11 2022-12-20 13:58:13 2023-05-18 13:33:16 0000-00-00 00:00:00 2022 0 0 Str. 2532-2553 iss. 6 Vol. 25 Dec. 2022 0000-00-00 PostprintKoncna Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 2023-12-01 517.956 1311-0454 10.1007/s13540-022-00105-4 130585603 RAZ_Zuo_Jiabin_2022.pdf RAZ_Zuo_Jiabin_2022.pdf 1 6DBA00D01C4856B36E704C002B78FED8 bd2843d42f31df24dc0be47addbc6ce2692500670288bfd6cee519450693381e b2cc5c40-ea57-11ed-a314-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=168615 https://link.springer.com/article/10.1007/s13540-022-00105-4 0 b8ea81be-8066-11ed-9c94-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=165126 Pedagoška fakulteta Fakulteta za matematiko in fiziko 0 0 0