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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 45–61 | Cite as

Convergence From Power-Law to Logarithm-Law in Nonlinear Scalar Field Equations

  • Zhi-Qiang WangEmail author
  • Chengxiang Zhang
Article
  • 110 Downloads

Abstract

In this note, we uncover a relation between power-law nonlinear scalar field equations and logarithmic-law scalar field equations.We show that the ground state solutions, as p\({\downarrow}\) 2 for the power-law scalar field equations, converge to the ground state solutions of the logarithmic-law equations. As an application of this relation, we show that the associated Sobolev inequalities for imbedding from W1,2(\({\mathbb{R}^{N}}\)) into Lp (\({\mathbb{R}^{N}}\)) converge to an associated logarithmic Sobolev inequality, giving a new proof of the latter inequality due to Lieb–Loss (Analysis, 2nd edn, Graduate studies in mathematics, 14, American Mathematical Society, Providence, 2001). Using this relation, we also derive a Liouville type theorem for positive solutions of the nonlinear scalar field equation with power-law nonlinearity, giving a sharp version of an earlier result in Felmer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 25(1): 105–119, 2008).

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References

  1. 1.
    Adams R.A., Clarke F.H.: Gross’s logarithmic Sobolev inequality: a simple proof. Am. J. Math. 101(6), 1265–1269 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bahri A., Lions P.L.: Morse index of some min-max critical points. I. Application to multiplicity results. Commun. Pure Appl. Math. 41(8), 1027–1037 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berestycki H, Lions P.L.: Nonlinear scalar field equations, I existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berestycki H, Lions P.L.: Nonlinear scalar field equations, II existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82(4), 347–375 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berezin, F.A., Shubin, M.: The Schrödinger Equation. Springer (2012) Google Scholar
  6. 6.
    Białynicki-Birula I., Mycielski J.: Wave equations with logarithmic nonlinearities. Bull. Acad. Pol. Sci. Cl 3(23), 461–466 (1975)MathSciNetGoogle Scholar
  7. 7.
    Białynicki-Birula I., Mycielski J.: Nonlinear wave mechanics. Ann. Phys. 100(1–2), 62–93 (1976)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Cazenave T., Lions P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    d’Avenia P., Montefusco E., Squassina M.: On the logarithmic Schrödinger equation. Commun. Contemp. Math. 16(2), 1350032 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Del Pino M., Dolbeault J.: The optimal Euclidean L p-Sobolev logarithmic inequality. J. Funct. Anal. 197(1), 151–161 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Felmer P.L., Quaas A., Tang M., Yu J.: Monotonicity properties for ground states of the scalar field equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 25(1), 105–119 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Floer A., Weinstein A.: Nonspreading wave pockets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer (2001)Google Scholar
  15. 15.
    Gross L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kwong M.K.: Uniqueness of positive solutions of Δu − u +u p = 0 in \(\mathbb{R}^{n}\). Arch. Ration. Mech. Anal.1053 243–266 (1989)CrossRefGoogle Scholar
  17. 17.
    Lieb E.H., Loss M.: Analysis Second edition. Graduate Studies in Mathematics 14. American Mathematical Society, Providence (2001)Google Scholar
  18. 18.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, p. 65 (1986)Google Scholar
  19. 19.
    Rabinowitz P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math.Phys. 43(4), 270–291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Serrin J., Tang M.: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J. 49(3), 897–923 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Simon B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7(3), 447–526 (1982)CrossRefzbMATHGoogle Scholar
  22. 22.
    Squassina M., Szulkin A.: Multiple solutions to logarithmic Schrödinger equations with periodic potential. Calc. Var. Partial Differ. Equ. 54(1), 585–597 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Stam A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2(2), 101–112 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Strauss W.A.: Existence of solitarywaves in higher dimensions.Commun.Math. Phys. 55(2), 149–162 (1977)ADSCrossRefGoogle Scholar
  25. 25.
    Tanaka, K., Zhang, C.: Multi-bump solutions for logarithmic Schrödinger equations. Calc. Var. Partial Differ. Equ. 56(33) (2017)Google Scholar
  26. 26.
    Troy W.C.: Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation. Arch. Ration. Mech. Anal. 222(3), 1581–1600 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Weissler F.B.: Logarithmic Sobolev inequalities for the heat-diffusion semigroup. Trans. Am. Math. Soc. 237, 255–269 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zloshchastiev K.G.: Logarithmic nonlinearity in theories of quantum gravity: origin of time and observational consequences. Grav. Cosmol. 16, 288–297 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Znojil M., Ruzicka F., Zloshchastiev K.G.: Schrödinger equations with logarithmic self-interactions: from antilinear \({\mathcal{PT}}\)-symmetry to the nonlinear coupling of channels. Symmetry 9(8), 165 (2017)  https://doi.org/10.3390/sym9080165 CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA
  3. 3.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

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