# ANNALES DE L'INSTITUT FOURIER

BMO solutions to quasilinear equations of $p$-Laplace type
Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1911-1939.

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

We also characterize BMO solutions to equations $-{\Delta }_{p}u=\sigma {u}^{q}+\mu$ in ${ℝ}^{n}$, $u\ge 0$, with $q>0$, where both $\mu$ and $\sigma$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators $\mathrm{div}\left(𝒜\left(x,\nabla ·\right)\right)$ in place of ${\Delta }_{p}$.

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

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

Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3485
Classification: 35J92, 42B37, 31B15, 42B35
Keywords: BMO spaces, Wolff potentials, $p$-Laplacian
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)
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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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