{"title":"A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms","authors":"Roberta Filippucci, Yuhua Sun, Yadong Zheng","doi":"10.1007/s11854-024-0341-4","DOIUrl":null,"url":null,"abstract":"<p>In this article we study local and global properties of positive solutions of − Δ<sub><i>m</i></sub><i>u</i> = ∣<i>u</i>∣<sup><i>p</i>−1</sup><i>u</i>+<i>M</i>∣∇<i>u</i>∣<sup><i>q</i></sup> in a domain Ω of ℝ<sup><i>N</i></sup>, with <i>m</i> > 1, <i>p, q</i> > 0 and <i>M</i> ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.</p>","PeriodicalId":502135,"journal":{"name":"Journal d'Analyse Mathématique","volume":"41 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal d'Analyse Mathématique","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11854-024-0341-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
In this article we study local and global properties of positive solutions of − Δmu = ∣u∣p−1u+M∣∇u∣q in a domain Ω of ℝN, with m > 1, p, q > 0 and M ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.
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