Skip to main content
SpringerLink
Log in

Existence and Nonexistence of Solutions for the Double Phase Problem

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The aim of this paper is to establish the existence and the nonexistence of solutions for double-phase problems depending on two parameters. This work improves and complements the existing ones in the literature. There seems to be no results on the nonexistence of solutions for double-phase problems.

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. Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. PDE 57, 1–48 (2018)

    Article  MathSciNet  Google Scholar 

  2. Baroni, P., Colombo, M., Mingione, G.: Harnack inequalities for double phase functionals. Nonlinear Anal. 121, 206–222 (2015)

    Article  MathSciNet  Google Scholar 

  3. Byun, S.S., Oh, J.: Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains. J. Diff. Equ. 263, 1643–1693 (2017)

    Article  MathSciNet  Google Scholar 

  4. Benkirane, A., Sidi El Vally, M.: Variational inequalities in Musielak–Orlicz–Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21, 787–811 (2014)

    Article  MathSciNet  Google Scholar 

  5. Chang, K.C.: Critical Point Theory and Applications. Shanghai Scientific and thchnology Press, Shanghai (1996)

  6. Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215, 443–496 (2015)

    Article  MathSciNet  Google Scholar 

  7. Colombo, M., Mingione, G.: Bounded minimisers of double phase variational integrals. Arch. Ration. Mech. Anal. 218, 219–273 (2015)

    Article  MathSciNet  Google Scholar 

  8. Cencelj, M., Radulescu, V.D., Dusan, D.: Repovs, Double phase problems with variable growth. Nonlinear Anal. 177, 270–287 (2018)

    Article  MathSciNet  Google Scholar 

  9. Colasuonno, F., Squassina, M.: Eigenvalues for double phase variational integrals. Ann. Mat. Pura Appl. 195, 1917–1959 (2016)

    Article  MathSciNet  Google Scholar 

  10. De Filippis, C., Mingione, G.: Manifold constrained non-uniformly elliptic problems. J. Geom. Anal. 30, 1661–1723 (2020)

    Article  MathSciNet  Google Scholar 

  11. De Filippis, C., Mingione, G.: A borderline case of Calderón–Zygmund estimates for nonuniformly elliptic problems. St. Petersburg Math. J. 31, 455–477 (2020)

    Article  MathSciNet  Google Scholar 

  12. Fan, X., Guan, C.X.: Uniform convexity of Musielak–Orlicz–Sobolev spaces and applications. Nonlinear Anal. 73, 163–175 (2010)

    Article  MathSciNet  Google Scholar 

  13. Ge, B., Chen, Z.Y.: Existence of infinitely many solutions for double phase problem with sign-changing potential. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat 113, 3185–3196 (2019)

  14. Ge, B., Lv, D.J., Lu, J.F.: Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions. Nonlinear Anal. 188, 294–315 (2019)

    Article  MathSciNet  Google Scholar 

  15. Ge, B., Wang, L.Y., Lu, J.F.: On a class of double-phase problem without Ambrosetti–Robinowitz-type conditions. Appl. Anal. (2019). https://doi.org/10.1080/00036811.2019.1679785

    Article  Google Scholar 

  16. Ge, B., Xue, X.P.: Multiple solutions for inequality Dirichlet problems by the \(p(x)\)-Laplacian. Nonlinear Anal. RWA 11, 3198–3210 (2010)

    Article  MathSciNet  Google Scholar 

  17. Hou, G.L., Ge, B., Zhang, B.L., Wang, L.Y.: Ground state sign-changing solutions for a class of double phase problem in bounded domains. Bound. Value Probl. 24, 1–21 (2020)

    MathSciNet  Google Scholar 

  18. Liu, W.L., Dai, G.W.: Existence and multiplicity results for double phase problem. J. Diff. Equ. 265, 4311–4334 (2018)

    Article  MathSciNet  Google Scholar 

  19. Liu, W.L., Dai, G.W.: Three ground state solutions for double phase problem. J. Math. Phys. 59, (2018)

  20. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math, vol. 1034. Springer, Berlin (1983)

  21. Mingione, G., Rădulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501, (2021)

  22. Papageorgiou, N.S., Rădulescu, V.D., Repovs, D.: Double-phase problems and a discontinuity property of the spectrum. Proc. Am. Math. Soc. 147, 2899–2910 (2019)

    Article  MathSciNet  Google Scholar 

  23. Papageorgiou, N.S., Rădulescu, V.D., Repovs, D.: Positive solutions for nonlinear Neumann problems with singular terms and convection. J. Math. Pures Appl. 136, 1–21 (2020)

    Article  MathSciNet  Google Scholar 

  24. Perera, K., Squassina, M.: Existence results for double-phase problems via Morse theory. Commun. Contemp. Math. 20, 1750023 (2018)

    Article  MathSciNet  Google Scholar 

  25. Ragusa, M., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9, 710–728 (2020)

    Article  MathSciNet  Google Scholar 

  26. Ricceri, B.: A further three critical points theorem. Nonlinear Anal. 71, 4151–4157 (2009)

    Article  MathSciNet  Google Scholar 

  27. Shi, X., Radulescu, V.D., Repovs, D.D., Zhang, Q.H.: Multiple solutions of double phase variational problems with variable exponent. Adv. Calc. Var. 13, 385–401 (2020)

    Article  MathSciNet  Google Scholar 

  28. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50, 675–710 (1986)

    MathSciNet  Google Scholar 

  29. Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)

    MathSciNet  MATH  Google Scholar 

  31. Zhang, Q., Radulescu, V.D.: Double phase anisotropic variational problems and combined effects of reaction and absorption terms. J. Math. Pures Appl. 118, 159–203 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, People’s Republic of China

    Xiao-Feng Cao, Bin Ge & Wen-Shou Yuan

Authors
  1. Xiao-Feng Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wen-Shou Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Ge.

Ethics declarations

Conflicts of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China (No. 11201095), the Fundamental Research Funds for the Central Universities of China (No. 3072020CF2401), the Postdoctoral research startup foundation of Heilongjiang (No. LBH-Q14044), the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (No. LC201502).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, XF., Ge, B. & Yuan, WS. Existence and Nonexistence of Solutions for the Double Phase Problem. Results Math 76, 132 (2021). https://doi.org/10.1007/s00025-021-01444-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-021-01444-z

Keywords

Mathematics Subject Classification

Search

Navigation