BMO solutions to quasilinear equations of $p$-Laplace type

Phuc, Nguyen Cong; Verbitsky, Igor E.

doi:10.5802/aif.3485

BMO solutions to quasilinear equations of

p

-Laplace type

Phuc, Nguyen Cong ¹; Verbitsky, Igor E.²

¹ Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803 (USA)
² Department of Mathematics, University of Missouri, Columbia, MO 65211 (USA)

Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1911-1939.

Abstract
Résumé

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $- Δ_{p} u = μ$ in $ℝ^{n}$ , $u \geq 0$ , where $μ$ is a locally finite Radon measure, and $Δ_{p} u = {div (| \nabla u |}^{p - 2} \nabla u)$ is the $p$ -Laplacian ( $p > 1$ ).

We also characterize BMO solutions to equations $- Δ_{p} u = σ u^{q} + μ$ in $ℝ^{n}$ , $u \geq 0$ , with $q > 0$ , where both $μ$ and $σ$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators $div (𝒜 (x, \nabla \cdot))$ in place of $Δ_{p}$ .

Nous donnons les conditions nécessaires et suffisantes pour l’existence d’une solution BMO de l’équation quasi-linéaire $- Δ_{p} u = μ$ dans $ℝ^{n}$ , $u \geq 0$ , où $μ$ est une mesure de Radon localement finie, et $Δ_{p} u = {div (| \nabla u |}^{p - 2} \nabla u)$ est le $p$ -Laplacien ( $p > 1$ ).

Nous caractérisons également les solutions BMO de l’équation $- Δ_{p} u = σ u^{q} + μ$ dans $ℝ^{n}$ , $u \geq 0$ , avec $q > 0$ , où $μ$ et $σ$ sont des mesures de Radon localement finies. Nos principaux résultats sont valables pour la classe plus générale des opérateurs quasi-linéaires plus généraux $div (𝒜 (x, \nabla \cdot))$ à la place de $Δ_{p}$ .

Received: 2021-01-19
Revised: 2021-04-09
Accepted: 2021-05-06
Published online: 2022-09-12

DOI: 10.5802/aif.3485

Classification: 35J92, 42B37, 31B15, 42B35
Keywords: BMO spaces, Wolff potentials,

p

-Laplacian

Author's affiliations:

Phuc, Nguyen Cong ¹; Verbitsky, Igor E. ²

¹ Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803 (USA)
² Department of Mathematics, University of Missouri, Columbia, MO 65211 (USA)

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Phuc, Nguyen Cong and Verbitsky, Igor E.},
     title = {BMO solutions to quasilinear equations of $p${-Laplace} type},
     journal = {Annales de l'Institut Fourier},
     pages = {1911--1939},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {5},
     year = {2022},
     doi = {10.5802/aif.3485},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3485/}
}

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Phuc, Nguyen Cong; Verbitsky, Igor E. BMO solutions to quasilinear equations of $p$-Laplace type. Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1911-1939. doi : 10.5802/aif.3485. https://aif.centre-mersenne.org/articles/10.5802/aif.3485/

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