Infinitely Many Solutions for p(x)-Laplacian-Like Neumann Problems with Indefinite Weight

  • Zhou Qing-MeiEmail author
  • Wang Ke-Qi


In the present paper, in view of the variational approach, we discuss the Neumann problems with indefinite weight and p(x)-Laplacian-like operators, originated from a capillary phenomena. Under certain assumptions, we prove the existence of infinitely many nontrival solutions of the problem.


p(x)-Laplacian-like Neumann problem variational method infinitely many solutions 

Mathematics Subject Classification

Primary 35D05 Secondary 35J60 35J70 



  1. 1.
    Diening, L., Hästö, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: Drábek, P., Rákosník, J. (eds.) FSDONA04 Proceedings, pp. 38–58. Milovy, Czech Republic (2004)Google Scholar
  2. 2.
    Rodrigues, M.M.: Multiplicity of solutions on a nonlinear eigenvalue problem for \(p(x)\)-Laplacian-like operators. Mediterr. J. Math. 9, 211–223 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ge, B.: On superlinear \(p(x)\)-Laplacian-like problem without Ambrosetti and Rabinowitz condition. Bull. Korean Math. Soc. 51, 409–421 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zhou, Q.M.: On the superlinear problems involving \(p(x)\)-Laplacian-like operators without AR-condition. Nonlinear Anal. RWA 12, 161–169 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Liu, D.C.: On a \(p\)-Kirchhoff equation via fountain theorem and dual fountain theorem. Nonlinear Anal. 72, 302–308 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dai, G.W., Wei, J.: Infinitely many non-negative solutions for a \(p(x)\)-Kirchhoff type problem with Dirichlet boundary condition. Nonlinear Anal. 73, 3420–3430 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cammaroto, F., Vilasi, L.: Multiplicity results for a Neumann boundary value problem involving the \(p(x)\)-Laplacian. Taiwan. J. Math. 16(2), 621–634 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cammaroto, F., Vilasi, L.: Multiple solutions for a Kirchhoff-type problem involving the \(p(x)\)-Laplacian operator. Nonlinear Anal. 74, 1841–1852 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Musielak, J.: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)CrossRefGoogle Scholar
  10. 10.
    Edmunds, D.E., Lang, J., Nekvinda, A.: On \(L^{p(x)}(\Omega )\) norms. Proc. R. Soc. Ser. A. 455, 219–225 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Edmunds, D.E., Rákosník, J.: Sobolev embedding with variable exponent. Stud. Math. 143, 267–293 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}(\Omega )\) and \(W^{k, p(x)}(\Omega )\). Czechoslov. Math. J. 41(4), 592–618 (1991)zbMATHGoogle Scholar
  13. 13.
    Fan, X.L., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces \(W^{1, p(x)}\). J. Math. Anal. Appl. 262(2), 749–760 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fan, X.L., Zhao, D.: On the space \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fan, X.L.: Eigenvalues of the \(p(x)\)-Laplacian Neumann Problems. Nonlinear Anal. 67, 2982–2992 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  17. 17.
    Willem, M.: Minimax Theorems. Birkhauser, Basel (1996)CrossRefGoogle Scholar
  18. 18.
    Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Library, Northeast Forestry UniversityHarbinPeople’s Republic of China
  2. 2.College of Mechanical and Electrical EngineeringNortheast Forestry UniversityHarbinPeople’s Republic of China

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