Uniform boundedness results of solutions to mixed local and nonlocal elliptic operator

Xicuo Zha; Shuibo Huang; Qiaoyu Tian; Xicuo Zha; Shuibo Huang; Qiaoyu Tian

doi:10.3934/math.20231053

AIMS Mathematics

2023, Volume 8, Issue 9: 20665-20678. doi: 10.3934/math.20231053

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Research article

Uniform boundedness results of solutions to mixed local and nonlocal elliptic operator

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

Received: 01 May 2023 Revised: 14 June 2023 Accepted: 16 June 2023 Published: 27 June 2023
MSC : 35J67, 35R11

In this paper, by the Stampacchia method, we consider the boundedness of positive solutions to the following mixed local and nonlocal quasilinear elliptic operator

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

where $ s\in(0, 1) $, $ 1 < p < N $, $ f\in L^{m}(\Omega) $ with $ m > \frac{Np}{p(s+p-1)-\gamma(N-sp)} $, $ 0\leqslant\gamma < p_s^*-1 $, $ p_s^{*} = \frac{Np}{N-sp} $ is the critical Sobolev exponent.
- mixed local and nonlocal operator,
- bioundedness,
- Stampacchia method
Citation: Xicuo Zha, Shuibo Huang, Qiaoyu Tian. Uniform boundedness results of solutions to mixed local and nonlocal elliptic operator[J]. AIMS Mathematics, 2023, 8(9): 20665-20678. doi: 10.3934/math.20231053

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Abstract

In this paper, by the Stampacchia method, we consider the boundedness of positive solutions to the following mixed local and nonlocal quasilinear elliptic operator

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

where $ s\in(0, 1) $, $ 1 < p < N $, $ f\in L^{m}(\Omega) $ with $ m > \frac{Np}{p(s+p-1)-\gamma(N-sp)} $, $ 0\leqslant\gamma < p_s^*-1 $, $ p_s^{*} = \frac{Np}{N-sp} $ is the critical Sobolev exponent.

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