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Critical points of the Moser–Trudinger functional on closed surfaces

Abstract

Given a closed Riemann surface \((\Sigma ,{g_0})\) and any positive weight \(f\in C^\infty (\Sigma )\), we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional

$$\begin{aligned} {I_{p,\beta }}(u)=\frac{2-p}{2}\left( \frac{p\Vert u\Vert _{H^1}^2}{2\beta } \right) ^{\frac{p}{2-p}}-\ln \int _\Sigma \left( e^{u_+^p}-1\right) {f}\, dv_{{g_0}}, \end{aligned}$$

for every \(p\in (1,2)\) and \(\beta >0\), or for \(p=1\) and \(\beta \in (0,\infty ){\setminus } 4\pi {\mathbb {N}}\). Letting \(p\uparrow 2\) we obtain positive critical points of the Moser-Trudinger functional

$$\begin{aligned} F(u):=\int _\Sigma \left( e^{u^2}-1\right) {f}\,dv_{{g_0}} \end{aligned}$$

constrained to \({\mathcal {E}}_\beta :=\left\{ v\text { s.t. }\Vert v\Vert _{H^1}^2=\beta \right\} \) for any \(\beta >0\).

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References

  1. Bartolucci, D., Lin, C.-S.: Existence and uniqueness for mean field equations on multiply connected domains at the critical parameter. Math. Ann. 359(1–2), 1–44 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  2. Battaglia, L., Jevnikar, A., Malchiodi, A., Ruiz, D.: A general existence result for the Toda system on compact surfaces. Adv. Math. 285, 937–979 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  3. Carleson, L., Chang, S.-Y.A.: On the existence of an extremal function for an inequality of. J. Moser. Bull. Sci. Math. 110(2), 113–127 (1986)

    MathSciNet  MATH  Google Scholar 

  4. Chang, S.-Y. A., Chen, C.-C., Lin, C.-S.: Extremal functions for a mean field equation in two dimension. In: Lectures on partial differential equations, volume 2 of New Stud. Adv. Math., pages 61–93. Int. Press, Somerville, MA (2003)

  5. Chen, C.-C., Lin, C.-S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Commun. Pure Appl. Math. 55(6), 728–771 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  6. Chen, C.-C., Lin, C.-S.: Topological degree for a mean field equation on Riemann surfaces. Commun. Pure Appl. Math. 56(12), 1667–1727 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  7. Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615–622 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  8. Chen, W.X., Li, C.: Prescribing Gaussian curvatures on surfaces with conical singularities. J. Geom. Anal. 1(4), 359–372 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  9. Costa, D.G., Tintarev, C.: Concentration profiles for the Trudinger–Moser functional are shaped like toy pyramids. J. Funct. Anal. 266(2), 676–692 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  10. De Marchis, F.: Multiplicity result for a scalar field equation on compact surfaces. Commun. Partial Differ. Equ. 33(10–12), 2208–2224 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  11. De Marchis, F.: Generic multiplicity for a scalar field equation on compact surfaces. J. Funct. Anal. 259(8), 2165–2192 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  12. De Marchis, F., Ianni, I., Pacella, F.: Asymptotic profile of positive solutions of Lane-Emden problems in dimension two. J. Fixed Point Theory Appl. 19(1), 889–916 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  13. del Pino, M., Musso, M., Ruf, B.: New solutions for Trudinger–Moser critical equations in \(\mathbb{R} ^2\). J. Funct. Anal. 258(2), 421–457 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  14. del Pino, M., Musso, M., Ruf, B.: Beyond the Trudinger–Moser supremum. Calc. Var. Partial Differ. Equ. 44(3–4), 543–576 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  15. Deng, S., Musso, M.: Bubbling solutions for an exponential nonlinearity in \(\mathbb{R} ^2\). J. Differ. Equ. 257(7), 2259–2302 (2014)

    MATH  Article  MathSciNet  Google Scholar 

  16. Ding, W., Jost, J., Li, J., Wang, G.: The differential equation \(\Delta u=8\pi -8\pi he^u\) on a compact Riemann surface. Asian J. Math. 1(2), 230–248 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  17. Ding, W., Jost, J., Li, J., Wang, G.: Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(5), 653–666 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  18. Djadli, Z.: Existence result for the mean field problem on riemann surfaces of all genuses. Commun. Contemp. Math. 10(2), 205–220 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  19. Djadli, Z., Malchiodi, A.: Existence of conformal metrics with constant q-curvature. Ann. Math. 168(3), 813–858 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  20. do Carmo, M. P.: Differential geometry of curves and surfaces. Prentice-Hall, Inc., Englewood Cliffs, N.J., (1976). Translated from the Portuguese

  21. Druet, O.: Multibumps analysis in dimension 2: quantification of blow-up levels. Duke Math. J. 132(2), 217–269 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  22. Druet, O., Malchiodi, A., Martinazzi, L., Thizy, P.-D.: Multi-bumps analysis for Trudinger-Moser nonlinearities II-Existence of solutions of high energies. In preparation

  23. Druet, O., Thizy, P.-D.: Multi-bump analysis for Trudinger-Moser nonlinearities IQuantification and location of concentration points. J. Eur. Math. Soc. (2020). https://doi.org/10.4171/JEMS/1002

    Article  MATH  Google Scholar 

  24. Figueroa, P., Musso, M.: Bubbling solutions for Moser-Trudinger type equations on compact Riemann surfaces. J. Funct. Anal. 275(10), 2684–2739 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  25. Flucher, M.: Extremal functions for the Trudinger-Moser inequality in \(2\) dimensions. Comment. Math. Helv. 67(3), 471–497 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  26. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comment. Math. Helv. 68(3), 415–454 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  27. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition

  28. Han, Q., Lin, F.: Elliptic partial differential equations. Courant Lecture Notes in Mathematics, vol. 1. New York University Courant Institute of Mathematical Sciences, New York (1997)

  29. Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics, vol. 5. New York University Courant Institute of Mathematical Sciences, New York (1999)

  30. Ibrahim, S., Masmoudi, N., Nakanishi, K., Sani, F.: Sharp threshold nonlinearity for maximizing the Trudinger–Moser inequalities. J. Funct. Anal. 278(1), 108302 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  31. Judovič, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Akad. Nauk SSSR 138, 805–808 (1961). ((in Russian))

    MathSciNet  Google Scholar 

  32. Lamm, T., Robert, F., Struwe, M.: The heat flow with a critical exponential nonlinearity. J. Funct. Anal. 257(9), 2951–2998 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  33. Laurain, P.: Concentration of \(CMC\) surfaces in a 3-manifold. Int. Math. Res. Not. IMRN 24, 5585–5649 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  34. Li, Y.: Harnack type inequality: the method of moving planes. Commun. Math. Phys. 200(2), 421–444 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  35. Li, Y.: Moser–Trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equ. 14(2), 163–192 (2001)

    MathSciNet  MATH  Google Scholar 

  36. Lin, C.-S., Lucia, M.: Uniqueness of solutions for a mean field equation on torus. J. Differ. Equ. 229(1), 172–185 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  37. Malchiodi, A.: Morse theory and a scalar field equation on compact surfaces. Adv. Differ. Equ. 13(11–12), 1109–1129 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Malchiodi, A.: Topological methods for an elliptic equation with exponential nonlinearities. Discrete Contin. Dyn. Syst. 21, 277–294 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  39. Malchiodi, A., Martinazzi, L.: Critical points of the Moser-Trudinger functional on a disk. J. Eur. Math. Soc. (JEMS) 16(5), 893–908 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  40. Mancini, G., Martinazzi, L.: The Moser-Trudinger inequality and its extremals on a disk via energy estimates. Calc. Var. Partial Differ. Equ. 56(4), 26 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  41. Mancini, G., Thizy, P.-D.: Glueing a peak to a non-zero limiting profile for a critical Moser-Trudinger equation. J. Math. Anal. Appl. 472(2), 1430–1457 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  42. Mancini, G., Thizy, P.-D.: Critical points of Moser-Trudinger type functionals: a general picture. (2020). In preparation

  43. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., 20:1077–1092 (1970/71)

  44. Nolasco, M., Tarantello, G.: On a sharp Sobolev-type inequality on two-dimensional compact manifolds. Arch. Ration. Mech. Anal. 145(2), 161–195 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  45. Pohožaev, S.I.: On the eigenfunctions of the equation \(\delta u+\lambda f(u)=0\). Dokl. Akad. Nauk SSSR 165, 36–39 (1965). ((in Russian))

    MathSciNet  Google Scholar 

  46. Struwe, M.: Critical points of embeddings of \(H^{1, n}_0\) into Orlicz spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(5), 425–464 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  47. Struwe, M.: The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160(1–2), 19–64 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  48. Struwe, M.: Variational methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, (2008). Applications to nonlinear partial differential equations and Hamiltonian systems

  49. Suzuki, T.: Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(4), 367–397 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  50. Taylor, M. E.: Partial differential equations I. Basic theory, volume 115 of Applied Mathematical Sciences. Springer, New York, second edition (2011)

  51. Thizy, P.-D.: When does a perturbed Moser-Trudinger inequality admit an extremal ? Anal. PDE 13(5), 1371–1415 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  52. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

    MathSciNet  MATH  Google Scholar 

  53. Yang, Y.: Quantization for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary. Calc. Var. Partial Differ. Equ. 53(3–4), 901–941 (2015)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Pierre-Damien Thizy.

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De Marchis, F., Malchiodi, A., Martinazzi, L. et al. Critical points of the Moser–Trudinger functional on closed surfaces. Invent. math. (2022). https://doi.org/10.1007/s00222-022-01142-9

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  • DOI: https://doi.org/10.1007/s00222-022-01142-9