Existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic equations

Labudan Suonan; Yonglin Xu; Labudan Suonan; Yonglin Xu

doi:10.3934/math.20231268

AIMS Mathematics

2023, Volume 8, Issue 10: 24862-24887. doi: 10.3934/math.20231268

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Research article

Existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic equations

Labudan Suonan ,
Yonglin Xu ^,

School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China

Received: 14 July 2023 Revised: 14 August 2023 Accepted: 15 August 2023 Published: 24 August 2023
MSC : 35J67, 35R11

In this paper, we consider the existence of positive solutions to mixed local and nonlocal singular quasilinear singular elliptic equations

$ \begin{align*} \left\{\begin{array}{rl} -\Delta_{\vec{p}}u(x)+\left(-\Delta\right)_{p}^{s}u(x) = \frac{f(x)}{u(x)^{\delta}}, &x\in\Omega, \\ u(x)>0, \; \; \; \; \; \; &x\in\Omega, \\ u(x) = 0, \; \; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{array} \right. \end{align*} $

where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^{N}(N > 2) $, $ -\Delta_{\vec{p}}u $ is an anisotropic $ p $-Laplace operator, $ \vec{p} = (p_{1}, p_{2}, ..., p_{N}) $ with $ 2\leq p_{1}\leq p_{2}\leq\cdot\cdot\cdot\leq p_{N} $, $ \left(-\Delta \right)_{p}^{s} $ is the fractional $ p $-Laplace operator. The major results shows the interplay between the summability of the datum $ f(x) $ and the power exponent $ \delta $ in singular nonlinearities.
- mixed local and nonlocal,
- anisotropic $ p $-Laplace equation,
- singular elliptic operator
Citation: Labudan Suonan, Yonglin Xu. Existence of solutions to mixed local and nonlocal anisotropic quasilinear singular elliptic equations[J]. AIMS Mathematics, 2023, 8(10): 24862-24887. doi: 10.3934/math.20231268

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Abstract

In this paper, we consider the existence of positive solutions to mixed local and nonlocal singular quasilinear singular elliptic equations

$ \begin{align*} \left\{\begin{array}{rl} -\Delta_{\vec{p}}u(x)+\left(-\Delta\right)_{p}^{s}u(x) = \frac{f(x)}{u(x)^{\delta}}, &x\in\Omega, \\ u(x)>0, \; \; \; \; \; \; &x\in\Omega, \\ u(x) = 0, \; \; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{array} \right. \end{align*} $

where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^{N}(N > 2) $, $ -\Delta_{\vec{p}}u $ is an anisotropic $ p $-Laplace operator, $ \vec{p} = (p_{1}, p_{2}, ..., p_{N}) $ with $ 2\leq p_{1}\leq p_{2}\leq\cdot\cdot\cdot\leq p_{N} $, $ \left(-\Delta \right)_{p}^{s} $ is the fractional $ p $-Laplace operator. The major results shows the interplay between the summability of the datum $ f(x) $ and the power exponent $ \delta $ in singular nonlinearities.

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