Skip to main content
SpringerLink
Log in

Quasilinear Equations Using a Linking Structure with Critical Nonlinearities

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alves C O, Wang Y, Shen Y. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differential Equations, 2015, 259: 318–343

    Article  MathSciNet  Google Scholar 

  2. Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problems on ℝN. Comm Part Diff Eq, 1995, 20: 1725–1741

    Article  MathSciNet  Google Scholar 

  3. de Bouard A, Hayashi N, Saut J C. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm Math Phys, 1997, 189: 73–105

    Article  MathSciNet  Google Scholar 

  4. Brull L, Lange H. Solitary waves for quasilinear Schrödinger equations. Expo Math, 1986, 4: 278–288

    MATH  Google Scholar 

  5. Chen X L, Sudan R N. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys Rev Lett, 1993, 70: 2082–2085

    Article  Google Scholar 

  6. Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal, 2004, 56: 213–226

    Article  MathSciNet  Google Scholar 

  7. Del Pino M, Felmer P. Local Mountain Pass for semilinear elliptic problems in unbounded domains. Calc Var Partial Differential Equations, 1996, 4: 121–137

    Article  MathSciNet  Google Scholar 

  8. Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth. J Differential Equations, 2015, 258: 115–147

    Article  MathSciNet  Google Scholar 

  9. Furtado M F, Silva E D, Silva M L. Quasilinear Schrödinger equations with asymptotically linear nonlinearities. Adv Nonlinear Stud, 2014, 14: 671–686

    Article  MathSciNet  Google Scholar 

  10. Furtado M F, Silva E D, Silva M L. Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin. Z Angew Math Phys, 2015, 66: 277–291

    Article  MathSciNet  Google Scholar 

  11. Hasse R W. A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z Phys, 1980, 37: 83–87

    MathSciNet  Google Scholar 

  12. Kosevich A M, Ivanov B, Kovalev A S. Magnetic solitons. Phys Rep, 1990, 194: 117–238

    Article  Google Scholar 

  13. Kurihura S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Japan, 1981, 50: 3263–3267

    Google Scholar 

  14. Landau L D, Lifschitz E M. Quantum Mechanics, Non-relativistic Theory. Institute of Physical Problems URSS, Academy of Sciences, 1958

  15. Li Q, Wu X. Existence, multiplicity and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth. J Math Phys, 2017, 58: 041501

    Article  MathSciNet  Google Scholar 

  16. Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1: 109–145; 223–283

    Article  MathSciNet  Google Scholar 

  17. Litvak A G, Sergeev A M. One dimensional collapse of plasma waves. JETP Lett, 1978, 27: 517–520

    Google Scholar 

  18. Liu J Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations I. Proc Amer Math Soc, 2002, 131: 441–448

    Article  Google Scholar 

  19. Liu J Q. Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations II. J Differential Equations, 2003, 187: 473–493

    Article  MathSciNet  Google Scholar 

  20. Liu J Q, Wang Y Q, Wang Z Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differential Equations, 2004, 29: 879–901

    Article  MathSciNet  Google Scholar 

  21. Liu X, Liu J, Wang Z Q. Ground states for quasilinear Schrödinger equations with critical growth. Calc Var Partial Differential Equations, 2013, 46: 641–669

    Article  MathSciNet  Google Scholar 

  22. Liu S, Zhou J. Standing waves for quasilinear Schrödinger equations with indefinite potentials. Journal of Differential Equations, 2018, 265: 3970–3987

    Article  MathSciNet  Google Scholar 

  23. Makhankov V G, Fedyanin V K. Non-linear effects in quasi-one-dimensional models of condensed matter theory. Physics Reports, 1984, 104: 1–86

    Article  MathSciNet  Google Scholar 

  24. Nakamura A. Damping and modification of exciton solitary waves. J Phys Soc Jpn, 1977, 42: 1824–1835

    Article  Google Scholar 

  25. do Ó J M, Severo U B. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Commun Pure Appl Anal, 2009, 8: 621–644

    Article  MathSciNet  Google Scholar 

  26. do Ó J M, Miyagaki O H, Soares S H. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations, 2010, 248: 722–744

    Article  MathSciNet  Google Scholar 

  27. Poppenberg M, Schmitt K, Wang Z Q. On the existence of soliton solutions to quasilinear Schrödinger equations. Calc Var Partial Differential Equations, 2002, 14: 329–344

    Article  MathSciNet  Google Scholar 

  28. Rabinowitz P. Minimax methods in critical point theory with applications to differential equations. CBMS Reg Conf Ser Math. Vol 65. Providence RI: Amer Math Soc, 1986

    Book  Google Scholar 

  29. Rabinowitz P. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43: 270–291

    Article  MathSciNet  Google Scholar 

  30. Schechter M, Tintarev K. Pairs of critical points produced by linking subsets with applications to semilinear elliptic problems. Bull Soc Math Belg Ser B, 1992, 44: 249–261

    MathSciNet  MATH  Google Scholar 

  31. Schechter M. Linking methods in critical point theory. Boston, MA: Birkhäuser Boston, Inc, 1999

    Book  Google Scholar 

  32. Schechter M. A variation of the mountain pass lemma and applications. J London Math Soc, 1991, 44(2): 491–502

    Article  MathSciNet  Google Scholar 

  33. Silva E A B. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal, 1991, 16: 455–477

    Article  MathSciNet  Google Scholar 

  34. Silva E A B, Vieira G F. Quasilinear asymptotically periodic Schrödinger equations with subcritical growth. Nonlinear Analysis, 2010, 72: 2935–2949

    Article  MathSciNet  Google Scholar 

  35. Silva E A B, Vieira G F. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Cal Var, 2010, 39: 1–33

    Article  Google Scholar 

  36. Silva E D, Silva J S. Quasilinear Schrödinger equations with nonlinearities interacting with high eigenvalues. J Math Phys, 2019, 60: 081504

    Article  MathSciNet  Google Scholar 

  37. Silva E D, Silva J S. Multiplicity of solutions for critical quasilinear Schrödinger equations using a linking structure. Discrete Continuous Dynamical Systems — A, 2020, 40(9): 5441–5470

    Article  Google Scholar 

  38. Souto M A S, Soares S H M. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Commun Pure Appl Anal, 2013, 12(1): 99–116

    Article  MathSciNet  Google Scholar 

  39. Willem M. Minimax Theorems. Basel, Berlin: Birkhäuser Boston, 1996

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Federal University of Goiás, Zip code 74001-970, Goiânia, Goiás-GO, Brazil

    Edcarlos D. Silva & Jefferson S. Silva

Authors
  1. Edcarlos D. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jefferson S. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edcarlos D. Silva.

Additional information

Research was partially supported by CNPq with (429955/2018-9). The first author was partially suported by CNPq (309026/2020-2) and FAPDF with (16809.78.45403.25042017).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Silva, E.D., Silva, J.S. Quasilinear Equations Using a Linking Structure with Critical Nonlinearities. Acta Math Sci 42, 975–1002 (2022). https://doi.org/10.1007/s10473-022-0310-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-022-0310-x

Key words

2010 MR Subject Classification

Search

Navigation