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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Acknowledgements
The authors would like to thank the anonymous referees for their careful reviews and helpful comments related to Theorems 1.2, 1.4 and 1.9.
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Filippucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and was partly supported by Fondo Ricerca di Base di Ateneo Esercizio 2017–19 of the University of Perugia “Problemi con non linearità dipendenti dal gradiente” and by GNAMPA-INdAM Project 2023 “Equazioni differenziali alle derivate parziali nella modellizzazione di fenomeni reali” (CUP_E53C22001930001)
Sun was supported by the National Natural Science Foundation of China (No. 12371206).
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Filippucci, R., Sun, Y. & Zheng, Y. A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms. JAMA (2024). https://doi.org/10.1007/s11854-024-0341-4
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DOI: https://doi.org/10.1007/s11854-024-0341-4