Abstract
In this paper, we use the Nash–Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation \(\Delta _pv +av^{q}=0\) defined on a complete Riemannian manifolds (M, g) where \(p>1\), a and q are constants and \(\Delta _p(v)=\textrm{div}(|\nabla v|^{p-2}\nabla v)\) is the p-Laplace operator. Under some assumptions on a, p and q, we derive gradient estimates and Liouville type theorems for such positive solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bidaut-Véron, M.-F.: Local and global behavior of solutions of quasilinear equations of Emden–Fowler type. Arch. Ration. Mech. Anal. 107(4), 293–324 (1989)
Bidaut-Véron, M.-F., Véron, L.: Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math. 106(3), 489–539 (1991)
Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42(3), 271–297 (1989)
Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615–622 (1991)
Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975)
Constantin, A., Crowdy, D.G., Krishnamurthy, V.S., Wheeler, M.H.: Stuart-type polar vortices on a rotating sphere. Discrete Contin. Dyn. Syst. 41(1), 201–215 (2021)
Constantin, A., Germain, P.: Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere. Arch. Ration. Mech. Anal. 245(1), 587–644 (2022)
DiBenedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983)
Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34(4), 525–598 (1981)
Grigor’yan, A., Sun, Y.: On nonnegative solutions of the inequality \(\Delta u+u^\sigma \le 0\) on Riemannian manifolds. Commun. Pure Appl. Math. 67(8), 1336–1352 (2014)
Huang, G., Guo, Q., Guo, L.: Gradient estimates for positive weak solution to \(\delta _pu+au^{\sigma }=0\) on Riemannian manifolds. arXiv preprint arXiv:2304.04357 (2023)
Kotschwar, B., Ni, L.: Local gradient estimates of \(p\)-harmonic functions, \(1/H\)-flow, and an entropy formula. Ann. Sci. Éc. Norm. Supér. (4) 42(1), 1–36 (2009)
Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)
Ni, W.-M., Serrin, J.: Nonexistence theorems for singular solutions of quasilinear partial differential equations. Commun. Pure Appl. Math. 39(3), 379–399 (1986)
Peng, B., Wang, Y., Wei, G.: Yau type gradient estimates for \(\Delta u + au (\log u)^p + bu = 0\) on Riemannian manifolds. J. Math. Anal. Appl. 498, 124963 (2021)
Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differ. Geom. 36(2), 417–450 (1992)
Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20(2), 479–495 (1984)
Schoen, R.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl. Math. 41(3), 317–392 (1988)
Schoen, R., Yau, S.T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math. 92(1), 47–71 (1988)
Serrin, J., Zou, H.: Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189(1), 79–142 (2002)
Sun, Y.: On nonexistence of positive solutions of quasi-linear inequality on Riemannian manifolds. Proc. Am. Math. Soc. 143(7), 2969–2984 (2015)
Sung, C.-J.A., Wang, J.: Sharp gradient estimate and spectral rigidity for \(p\)-Laplacian. Math. Res. Lett. 21(4), 885–904 (2014)
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51(1), 126–150 (1984)
Uhlenbeck, K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138(3–4), 219–240 (1977)
Wang, X., Zhang, L.: Local gradient estimate for \(p\)-harmonic functions on Riemannian manifolds. Commun. Anal. Geom. 19(4), 759–771 (2011)
Wang, Y., Wei, G.: On the nonexistence of positive solution to \(\Delta u + au^{p+1} = 0\) on Riemannian manifolds. J. Differ. Equ. 362, 74–87 (2023)
Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
Zhao, L., Yang, D.: Gradient estimates for the \(p\)-Laplacian Lichnerowicz equation on smooth metric measure spaces. Proc. Am. Math. Soc. 146(12), 5451–5461 (2018)
Acknowledgements
The authors are grateful to the referee for valuable suggestions and insightful comments. The author Y. Wang is supported partially by NSFC (Grant No.11971400) and National key Research and Development projects of China (Grant No. 2020YFA0712500). G. Wei is supported by National Natural Science Foundation of China (Grants No.12101619 and 12141106).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
He, J., Wang, Y. & Wei, G. Gradient estimate for solutions of the equation \(\Delta _pv +av^{q}=0\) on a complete Riemannian manifold. Math. Z. 306, 42 (2024). https://doi.org/10.1007/s00209-024-03446-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00209-024-03446-3