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Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds (2311.02568v2)

Published 5 Nov 2023 in math.AP and math.DG

Abstract: In this paper, we provide a new routine to employ the Nash-Moser iteration technique to analyze the local and global properties of positive solutions to the equation $$\Delta_pv + a|\nabla v|qvr =0$$ on a complete Riemannian manifold with Ricci curvature bounded from below, where $p>1$, $q$, $r$ and $a$ are some real constants. Assuming certain conditions on $a,\, p,\, q$ and $r$, we can derive universal and succinct Cheng-Yau type logarithmic gradient estimates for such solutions. In particular, we give the obvious expressions of constants in the logarithmic gradient estimate for entire solutions to the above equation (see \thmref{t10}). The gradient estimates enable us to obtain some Liouville-type theorems, Harnack inequalities and some local estimates near singularities for positive solutions. Some of our results are new even in the case the domain is an Euclidean space and $p=2$.

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References (50)
  1. M.-F. Bidaut-Véron. Local and global behavior of solutions of quasilinear equations of Emden-Fowler type. Arch. Rational Mech. Anal., 107(4):293–324, 1989.
  2. Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math., 84 (2001), 1–49.
  3. Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. Invent. Math., 106(3):489–539, 1991.
  4. Local and global properties of solutions of quasilinear Hamilton-Jacobi equations. J. Funct. Anal., 267(9):3294–3331, 2014.
  5. Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient. Duke Math. J., 168(8):1487–1537, 2019.
  6. M.-F. Bidaut-Véron. Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term. Adv. Nonlinear Stud., 21(1):57–76, 2021.
  7. H. Brézis and P. L. Lions. “A note on isolated singularities for linear elliptc equations” in Mathematical Analysis and Applications, Part A, Academic Press, New York, 1981, 263-266.
  8. L. Caffarelli, N. Garofalo and F. Segala. A gradient bound for entire solutions of quasi-linear equations and its consequences. Comm. Pure Appl. Math., 47 (1994), 1457-1473.
  9. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math., 42(3):271–297, 1989.
  10. Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms. Nonlinear Anal., 220:Paper No. 112873, 29, 2022.
  11. Classification of solutions of some nonlinear elliptic equations. Duke Math. J., 63(3):615–622, 1991.
  12. S. Y. Cheng and S. T. Yau. Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354.
  13. T. H. Colding and W. P. Minicozzi II. Liouville properties. ICCM Not. 7 (2019), no. 1, 16-26.
  14. T. H. Colding and W. P. Minicozzi II. Harmonic functions on manifolds. Ann. of Math. (2) 146 (1997), no. 3, 725-747.
  15. T. H. Colding and W. P. Minicozzi II, Weyl type bounds for harmonic functions. Inventiones Math. 131 (1998), 257-298.
  16. T. H. Colding and W. P. Minicozzi II, Liouville theorems for harmonic sections and applications. Comm. Pure Appl. Math. 52 (1998), 113-138.
  17. E. N. Dancer. Superlinear problems on domains with holes of asymptotic shape and exterior problems. Math. Z. 229 (1998), no. 3, 475–491.
  18. E. Di Benedetto. C1+αsuperscript𝐶1𝛼C^{1+\alpha}italic_C start_POSTSUPERSCRIPT 1 + italic_α end_POSTSUPERSCRIPT local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal., 7(8):827–850, 1983.
  19. R. Filippucci. Nonexistence of positive weak solutions of elliptic inequalities. Nonlinear Anal., 70(8):2903–2916, 2009.
  20. R. Filippucci. Nonexistence of nonnegative solutions of elliptic systems of divergence type. J. Differential Equations, 250(1):572–595, 2011.
  21. R. Filippucci. Quasilinear elliptic systems in ℝNsuperscriptℝ𝑁\mathbb{R}^{N}blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT with multipower forcing terms depending on the gradient. J. Differential Equations, 255(7):1839–1866, 2013.
  22. A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient. Adv. Nonlinear Stud., 20(2):245–251, 2020.
  23. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math., 34(4):525–598, 1981.
  24. Gradient estimates for Δp⁢u−|∇u|q+b⁢(x)⁢|u|r−1⁢u=0subscriptΔ𝑝𝑢superscript∇𝑢𝑞𝑏𝑥superscript𝑢𝑟1𝑢0\Delta_{p}u-|\nabla u|^{q}+b(x)|u|^{r-1}u=0roman_Δ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_u - | ∇ italic_u | start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT + italic_b ( italic_x ) | italic_u | start_POSTSUPERSCRIPT italic_r - 1 end_POSTSUPERSCRIPT italic_u = 0 on a complete Riemannian manifold and Liouville type theorems, preprint, 2023.
  25. Gradient estimate for solutions of the equation Δp⁢v+a⁢vq=0subscriptΔ𝑝𝑣𝑎superscript𝑣𝑞0\Delta_{p}v+av^{q}=0roman_Δ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_v + italic_a italic_v start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT = 0 on a complete riemannian manifold. Math. Z., 306(3):Paper No. 46, 19, 2024.
  26. Local gradient estimates of p𝑝pitalic_p-harmonic functions, 1/H1𝐻1/H1 / italic_H-flow, and an entropy formula. Ann. Sci. Éc. Norm. Supér. (4), 42(1):1–36, 2009.
  27. On the parabolic kernel of the Schrödinger operator. Acta Math., 156(3-4):153–201, 1986.
  28. Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90(2003), 27-87.
  29. P.-L. Lions. Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre. J. Analyse Math., 45:234–254, 1985.
  30. A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova, 234: 1–384, 2001. English translation in Proc. Steklov Inst. Math. no. 3, 1–362, 2001.
  31. Nonexistence theorems for singular solutions of quasilinear partial differential equations. Comm. Pure Appl. Math., 39(3):379–399, 1986.
  32. P. Polácˇˇ𝑐\check{c}overroman_ˇ start_ARG italic_c end_ARGik, P. Quittner and P. Souplet Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: elliptic equations and systems, Duke Math. J., 139(3), 555–574, 2007.
  33. L. Saloff-Coste. Uniformly elliptic operators on riemannian manifolds. Journal of Differential Geometry, 36(2):417–450, 1992.
  34. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei-Yue Ding, Kung-Ching Chang [Gong-Qing Zhang], Jia-Qing Zhong and Yi-Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, With a preface translated from the Chinese by Kaising Tso.
  35. J. Serrin. Local behavior of solutions of quasi-linear equations. Acta Math., 111(1964), 247–302.
  36. J. Serrin. Isolated singularities of solutions of quasi-linear equations. Acta Math., 113(1965), 219–240.
  37. Entire solutions of nonlinear Poisson equations. Proc. London Math. Soc., 24 (1972), 348–366.
  38. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math., 189(1):79–142, 2002.
  39. P. Souplet. The proof of the Lane-Emden conjecture in four space dimensions, Advances in Mathematics, 221:1409–-1427, (2009).
  40. A sharp Liouville principle for Δm⁢u+up⁢|∇u|q≤0subscriptΔ𝑚𝑢superscript𝑢𝑝superscript∇𝑢𝑞0\Delta_{m}u+u^{p}|\nabla u|^{q}\leq 0roman_Δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_u + italic_u start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT | ∇ italic_u | start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ≤ 0 on geodesically complete noncompact Riemannian manifolds. Math. Ann., 384(3-4):1309–1341, 2022.
  41. Sharp gradient estimate and spectral rigidity for p𝑝pitalic_p-Laplacian. Math. Res. Lett., 21(4):885–904, 2014.
  42. P. Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51(1):126–150, 1984.
  43. K. Uhlenbeck. Regularity for a class of non-linear elliptic systems. Acta Math., 138(3-4):219–240, 1977.
  44. You-De Wang. Harmonic maps from noncompact Riemannian manifolds with non-negative Ricci curvature outside a compact set. Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), no. 6, 1259-1275.
  45. Local gradient estimate for p𝑝pitalic_p-harmonic functions on Riemannian manifolds. Comm. Anal. Geom., 19(4):759–771, 2011.
  46. On the nonexistence of positive solution to Δ⁢u+a⁢up+1=0Δ𝑢𝑎superscript𝑢𝑝10\Delta u+au^{p+1}=0roman_Δ italic_u + italic_a italic_u start_POSTSUPERSCRIPT italic_p + 1 end_POSTSUPERSCRIPT = 0 on Riemannian manifolds. J. Differential Equations, 362:74–87, 2023.
  47. Gradient estimates for a parabolic p𝑝pitalic_p-Laplace equation with logarithmic nonlinearity on Riemannian manifolds. Proc. Amer. Math. Soc., 149(3):1329–1341, 2021.
  48. S.-T. Yau. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math., 28: 201–228, 1975.
  49. L. Zhao. Liouville theorem for weighted p𝑝pitalic_p-Lichnerowicz equation on smooth metric measure space. J. Differential Equations, 266(9): 5615–5624, 2019.
  50. Gradient estimates for the p𝑝pitalic_p-Laplacian Lichnerowicz equation on smooth metric measure spaces. Proc. Amer. Math. Soc., 146(12): 5451–5461, 2018.
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