We establish a local gradient estimate for positive solutions of the nonlinear elliptic equation $ \Delta u + f(u) = 0 $ on complete Riemannian manifolds with Ricci curvature bounded below. The proof relies on a new $ P $-function and the Moser iteration technique. As its applications, we prove new Liouville theorems that extend several known results.
Citation: Fan Chen, Jingxia Huang, Qihua Ruan. Local and global properties of positive solutions for $ \Delta u+f(u) = 0 $ on Riemannian manifolds[J]. AIMS Mathematics, 2026, 11(4): 11012-11030. doi: 10.3934/math.2026452
We establish a local gradient estimate for positive solutions of the nonlinear elliptic equation $ \Delta u + f(u) = 0 $ on complete Riemannian manifolds with Ricci curvature bounded below. The proof relies on a new $ P $-function and the Moser iteration technique. As its applications, we prove new Liouville theorems that extend several known results.
| [1] |
S. Bernstein, Sur la généralisation du problème de Dirichlet, Math. Ann., 69 (1910), 82–136. https://doi.org/10.1007/BF01455154 doi: 10.1007/BF01455154
|
| [2] |
M. F. Bidaut-Veron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293–324. https://doi.org/10.1007/BF00251552 doi: 10.1007/BF00251552
|
| [3] |
M. F. Bidaut-Veron, L. Veron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489–539. https://doi.org/10.1007/BF01243922 doi: 10.1007/BF01243922
|
| [4] |
L. A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271–297. https://doi.org/10.1002/cpa.3160420304 doi: 10.1002/cpa.3160420304
|
| [5] |
S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., 28 (1975), 333–354. https://doi.org/10.1002/cpa.3160280303 doi: 10.1002/cpa.3160280303
|
| [6] |
W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615–622. https://doi.org/10.1215/S0012-7094-91-06325-8 doi: 10.1215/S0012-7094-91-06325-8
|
| [7] |
A. Constantin, D. G. Crowdy, V. S. Krishnamurthy, M. H. Wheeler, Stuart-type polar vortices on a rotating sphere, Discrete Contin. Dyn. Syst., 41 (2021), 201–215. https://doi.org/10.3934/dcds.2020263 doi: 10.3934/dcds.2020263
|
| [8] |
A. Constantin, D. P. Germain, Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere, Arch. Rational Mech. Anal., 245 (2022), 587–644. https://doi.org/10.1007/s00205-022-01791-3 doi: 10.1007/s00205-022-01791-3
|
| [9] |
A. Constantin, V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 865 (2019), 1072–1084. https://doi.org/10.1017/jfm.2019.109 doi: 10.1017/jfm.2019.109
|
| [10] |
F. Cuccu, A. Mohammed, G. Porru, Extensions of a theorem of Cauchy-Liouville, J. Math. Anal. Appl., 369 (2011), 222–231. https://doi.org/10.1016/j.jmaa.2010.02.058 doi: 10.1016/j.jmaa.2010.02.058
|
| [11] |
E. N. Dancer, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z., 229 (1998), 475–491. https://doi.org/10.1007/PL00004666 doi: 10.1007/PL00004666
|
| [12] | C. Enache, A Liouville type theorem for $p$-laplace equations, Electron. J. Diff. Equ., 2015 (2015), 1–7. |
| [13] |
B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525–598. https://doi.org/10.1002/cpa.3160340406 doi: 10.1002/cpa.3160340406
|
| [14] | C. Guo, Z. Zhang, A New proof of Liouville type theorems for a class of semilinear elliptic equations, arXiv, 2025. https://doi.org/10.48550/arXiv.2507.07664 |
| [15] | J. He, L. Sun, Y. Wang, Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds, arXiv, 2025. https://doi.org/10.48550/arXiv.2411.06956 |
| [16] |
J. He, Y. Wang, G. Wei, Gradient estimate for solutions of the equation $\Delta_{p}v+av^{q} = 0$ on a complete Riemannian manifold, Math. Z., 306 (2024), 42. https://doi.org/10.1007/s00209-024-03446-3 doi: 10.1007/s00209-024-03446-3
|
| [17] |
B. Kotschwar, L. Ni, Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula, Ann. Sci. l'École Norm. Supér., 42 (2009), 1–36. https://doi.org/10.24033/asens.2089 doi: 10.24033/asens.2089
|
| [18] |
J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., 100 (1991), 233–256. https://doi.org/10.1016/0022-1236(91)90110-Q doi: 10.1016/0022-1236(91)90110-Q
|
| [19] |
Y. Li, L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27–87. https://doi.org/10.1007/BF02786551 doi: 10.1007/BF02786551
|
| [20] |
D. Lin, X. Ma, Liouville type theorem for a class quasilinear $p$-Laplace type equation on the sphere, Math. Ann., 389 (2024), 2273–2287. https://doi.org/10.1007/s00208-023-02709-4 doi: 10.1007/s00208-023-02709-4
|
| [21] |
Z. Lu, Liouville theorems and Harnack inequality for a class of semilinear elliptic equations, J. Differ. Equations, 426 (2025), 454–465. https://doi.org/10.1016/j.jde.2025.01.068 doi: 10.1016/j.jde.2025.01.068
|
| [22] | Z. Lu, Logarithmic gradient estimate and universal bounds for semilinear elliptic equations revisited, arXiv, 2023. https://doi.org/10.48550/arXiv.2308.14026 |
| [23] |
Z. Lu, Some remarks for semilinear elliptic equations on Riemannian manifolds with nonnegative curvature, Ann. Mat. Pure Appl., 204 (2025), 2377–2390. https://doi.org/10.1007/s10231-025-01575-y doi: 10.1007/s10231-025-01575-y
|
| [24] |
X. Ma, Q. Ou, A Liouville theorem for a class semilinear elliptic equations on the Heisenberg group, Adv. Math., 413 (2023), 108851. https://doi.org/10.1016/j.aim.2022.108851 doi: 10.1016/j.aim.2022.108851
|
| [25] |
J. A. McCoy, Bernstein properties of solutions to some higher-order equations, Differ. Integral Equ., 20 (2007), 1153–1166. https://doi.org/10.57262/die/1356039300 doi: 10.57262/die/1356039300
|
| [26] |
W. M. Ni, J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations, Commun. Pure Appl. Math., 39 (1986), 379–399. https://doi.org/10.1002/cpa.3160390306 doi: 10.1002/cpa.3160390306
|
| [27] |
P. Pol$\acute{a}\check{c}$ik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555–579. https://doi.org/10.1215/S0012-7094-07-13935-8 doi: 10.1215/S0012-7094-07-13935-8
|
| [28] |
P. Quittner, Optimal Liouville theorems for superlinear parabolic problems, Duke Math. J., 170 (2021), 1113–1136. https://doi.org/10.1215/00127094-2020-0096 doi: 10.1215/00127094-2020-0096
|
| [29] |
J. Serrin, H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79–142. https://doi.org/10.1007/BF02392645 doi: 10.1007/BF02392645
|
| [30] |
L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differ. Geom., 36 (1992), 417–450. https://doi.org/10.4310/jdg/1214448748 doi: 10.4310/jdg/1214448748
|
| [31] |
X. Wang, L. Zhang, Local gradient estimate for $p$-harmonic functions on Riemannian manifolds, Commun. Anal. Geom., 19 (2011), 759–771. https://doi.org/10.4310/CAG.2011.v19.n4.a4 doi: 10.4310/CAG.2011.v19.n4.a4
|
| [32] |
Y. Wang, G. Wei, On the nonexistence of positive solution to $\Delta u+au^{p+1} = 0$ on Riemannian manifolds, J. Differ. Equations, 362 (2023), 74–87. https://doi.org/10.1016/j.jde.2023.03.001 doi: 10.1016/j.jde.2023.03.001
|
Fan Chen, Jingxia Huang, Qihua Ruan. Local and global properties of positive solutions for $ \Delta u+f(u) = 0 $ on Riemannian manifolds[J]. AIMS Mathematics, 2026, 11(4): 11012-11030. doi: 10.3934/math.2026452
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