Existence of solutions to elliptic equation with mixed local and nonlocal operators

Xiangrui Li; Shuibo Huang; Meirong Wu; Canyun Huang; Xiangrui Li; Shuibo Huang; Meirong Wu; Canyun Huang

doi:10.3934/math.2022735

AIMS Mathematics

2022, Volume 7, Issue 7: 13313-13324. doi: 10.3934/math.2022735

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Research article

Existence of solutions to elliptic equation with mixed local and nonlocal operators

1.
School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu, 730030, China
2.
The Department of Basic Courses, Army Logistics University, Chongqing, 430030, China
3.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received: 06 April 2022 Revised: 01 May 2022 Accepted: 05 May 2022 Published: 16 May 2022
MSC : 35J67, 35R11

In this paper, making use of a new non-smooth variational approach established by Moameni^{[13,14,15,16]}, we establish the existence of solutions to the following mixed local and nonlocal elliptic problem

$ \begin{equation*} \begin{cases} -\Delta u+(-\Delta)^s u = \mu g(x,u)+b(x), &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u\geq0,\; \; \; \; \; &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u = 0,\; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{cases} \end{equation*} $

where $ \Omega \subset \mathbb{R}^{N} $ is a bounded smooth domain, $ (-\Delta)^{s} $ is the restricted fractional Laplacian, $ \mu > 0 $, $ 0 < s < 1 $, $ N > 2s $, $ g $ satisfies some growth condition and $ b(x)\in L^m(\Omega) $ for $ m\geq 2 $. The interesting feature of our work is that we show that the nonlocal operator has an important influence in the existence of solutions to the above equation since $ g $ has new growth condition.
- non-smooth variational approach,
- mixed local and nonlocal operators
Citation: Xiangrui Li, Shuibo Huang, Meirong Wu, Canyun Huang. Existence of solutions to elliptic equation with mixed local and nonlocal operators[J]. AIMS Mathematics, 2022, 7(7): 13313-13324. doi: 10.3934/math.2022735

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Abstract

In this paper, making use of a new non-smooth variational approach established by Moameni^{[13,14,15,16]}, we establish the existence of solutions to the following mixed local and nonlocal elliptic problem

$ \begin{equation*} \begin{cases} -\Delta u+(-\Delta)^s u = \mu g(x,u)+b(x), &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u\geq0,\; \; \; \; \; &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u = 0,\; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{cases} \end{equation*} $

where $ \Omega \subset \mathbb{R}^{N} $ is a bounded smooth domain, $ (-\Delta)^{s} $ is the restricted fractional Laplacian, $ \mu > 0 $, $ 0 < s < 1 $, $ N > 2s $, $ g $ satisfies some growth condition and $ b(x)\in L^m(\Omega) $ for $ m\geq 2 $. The interesting feature of our work is that we show that the nonlocal operator has an important influence in the existence of solutions to the above equation since $ g $ has new growth condition.

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