Mathematische Annalen

, Volume 374, Issue 1–2, pp 67–98 | Cite as

Good-\(\lambda \) and Muckenhoupt–Wheeden type bounds in quasilinear measure datum problems, with applications

  • Quoc-Hung NguyenEmail author
  • Nguyen Cong Phuc


Weighted good-\(\lambda \) type inequalities and Muckenhoupt–Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in \(\mathbb {R}^n\) in the sense of Reifenberg. The principal operator here is modeled after the p-Laplacian, where for the first time singular case \(\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}\) is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence results and sharp bounds on the size of removable singular sets are deduced for a quasilinear Riccati type equation having a gradient source term with linear or super-linear power growth.

Mathematics Subject Classification

Primary 35J60 35J61 35J62 Secondary 35J75 42B37 



The authors kindly thank the anonymous referee for his/her comments that help improve the quality of the paper. Q.-H. Nguyen is supported by the Centro De Giorgi, Scuola Normale Superiore, Pisa, Italy. N. C. Phuc is supported in part by Simons Foundation, award number 426071.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Scuola Normale Superiore, Centro Ennio de GiorgiPisaItaly
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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