Existence of solutions for the fractional $ p＆q $-Laplacian equation with nonlocal Choquard reaction

Liyan Wang; Baocheng Zhang; Zhihui Lv; Kun Chi; Bin Ge; Liyan Wang; Baocheng Zhang; Zhihui Lv; Kun Chi; Bin Ge

doi:10.3934/math.2025416

AIMS Mathematics

2025, Volume 10, Issue 4: 9042-9054. doi: 10.3934/math.2025416

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Existence of solutions for the fractional $ p＆q $-Laplacian equation with nonlocal Choquard reaction

1.
School of Computer Science and Technology, Harbin University of Science and Technology, Harbin 150080, China
2.
Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China
3.
National Key Laboratory of Coal Coke and Graphite Testing, Harbin Costoms Technical Center, Hegang 154100, China
4.
School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China
5.
School of Automation, Harbin University of Science and Technology, Harbin 150080, China
6.
College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China

Received: 22 December 2024 Revised: 30 March 2025 Accepted: 09 April 2025 Published: 21 April 2025
MSC : 35J92, 35D05

We consider the following class of fractional $ p＆q $-Laplacian differential equation with Choquard term:
$ \left\{ \begin{array}{ll} (-\Delta )_{p}^{s}u+(-\Delta )_{q}^{s}u+V(x)(|u{{|}^{p-2}}u+|u{{|}^{q-2}}u) +\int_{{{\mathbb{R}}^{N}}}{g}(x)|u{{|}^{r}}dx = \int_{{{\mathbb{R}}^{N}}}{\int_{{{\mathbb{R}}^{N}}}{\frac{k(u(x))K(u(y))}{|x-y{{|}^{\alpha }}}}}dxdy,&\;x\in {{\mathbb{R}}^{N}}, \\ u\in W_{V}^{s,p}({{\mathbb{R}}^{N}})\bigcap{W_{V}^{s,q}}({{\mathbb{R}}^{N}}),&\;x\in {{\mathbb{R}}^{N}}, \end{array} \right. $
where $ s\in (0, 1), 2\le p\le r\le q < N/s, 0 < \alpha < N $, $ (-\Delta)_{m}^{s} $ with $ m \in \{p, q\} $ is the fractional $ m $-Laplacian operator, $ g(x):{{\mathbb{R}}^{N}}\to \mathbb{R} $, by introducing a potential term function to restore compactness in the corresponding spaces. Using variational techniques and inequalities such as Hardy–Littlewood–Sobolev, we ensure the geometric conditions of the mountain pass theorem in order to show the existence of solutions.
- fractional $ p＆q $-Laplacian,
- Hardy–Littlewood–Sobolev inequality,
- mountain pass theorem
Citation: Liyan Wang, Baocheng Zhang, Zhihui Lv, Kun Chi, Bin Ge. Existence of solutions for the fractional $ p＆q $-Laplacian equation with nonlocal Choquard reaction[J]. AIMS Mathematics, 2025, 10(4): 9042-9054. doi: 10.3934/math.2025416

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Abstract

We consider the following class of fractional $ p＆q $-Laplacian differential equation with Choquard term:

$ \left\{ \begin{array}{ll} (-\Delta )_{p}^{s}u+(-\Delta )_{q}^{s}u+V(x)(|u{{|}^{p-2}}u+|u{{|}^{q-2}}u) +\int_{{{\mathbb{R}}^{N}}}{g}(x)|u{{|}^{r}}dx = \int_{{{\mathbb{R}}^{N}}}{\int_{{{\mathbb{R}}^{N}}}{\frac{k(u(x))K(u(y))}{|x-y{{|}^{\alpha }}}}}dxdy,&\;x\in {{\mathbb{R}}^{N}}, \\ u\in W_{V}^{s,p}({{\mathbb{R}}^{N}})\bigcap{W_{V}^{s,q}}({{\mathbb{R}}^{N}}),&\;x\in {{\mathbb{R}}^{N}}, \end{array} \right. $

where $ s\in (0, 1), 2\le p\le r\le q < N/s, 0 < \alpha < N $, $ (-\Delta)_{m}^{s} $ with $ m \in \{p, q\} $ is the fractional $ m $-Laplacian operator, $ g(x):{{\mathbb{R}}^{N}}\to \mathbb{R} $, by introducing a potential term function to restore compactness in the corresponding spaces. Using variational techniques and inequalities such as Hardy–Littlewood–Sobolev, we ensure the geometric conditions of the mountain pass theorem in order to show the existence of solutions.

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