Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators

CaiDan LaMao; Shuibo Huang; Qiaoyu Tian; Canyun Huang; CaiDan LaMao; Shuibo Huang; Qiaoyu Tian; Canyun Huang

doi:10.3934/math.2022233

AIMS Mathematics

2022, Volume 7, Issue 3: 4199-4210. doi: 10.3934/math.2022233

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Research article

Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators

1.
School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu 730030, China
2.
Key Laboratory of Streaming Data Computing Technologies and Application, Northwest Minzu University, Lanzhou, Gansu 730030, China
3.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received: 07 November 2021 Revised: 02 December 2021 Accepted: 06 December 2021 Published: 17 December 2021
MSC : 35J67, 35R11

In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators

$ \left\{ \begin{matrix} - \Delta u(x)+{{(-\Delta )}^{s}}u(x)=f(x), & x\in \Omega , \\ u(x)\ge 0,~~~~~ & x\in \Omega , \\ u(x)=0,~~~~~ & x\in {{\mathbb{R}}^{N}}\setminus \Omega , \\ \end{matrix} \right. $

where $ 0 < s < 1 $, $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $ (-\Delta)^s $ is the fractional Laplace operator, $ f $ is a measurable function.
- fractional elliptic equations,
- mixed local and nonlocal operators,
- summability
Citation: CaiDan LaMao, Shuibo Huang, Qiaoyu Tian, Canyun Huang. Regularity results of solutions to elliptic equations involving mixed local and nonlocal operators[J]. AIMS Mathematics, 2022, 7(3): 4199-4210. doi: 10.3934/math.2022233

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Abstract

In this paper, we study the summability of solutions to the following semilinear elliptic equations involving mixed local and nonlocal operators

$ \left\{ \begin{matrix} - \Delta u(x)+{{(-\Delta )}^{s}}u(x)=f(x), & x\in \Omega , \\ u(x)\ge 0,~~~~~ & x\in \Omega , \\ u(x)=0,~~~~~ & x\in {{\mathbb{R}}^{N}}\setminus \Omega , \\ \end{matrix} \right. $

where $ 0 < s < 1 $, $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $ (-\Delta)^s $ is the fractional Laplace operator, $ f $ is a measurable function.

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