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Existence of three solutions for quasilinear elliptic equations: an Orlicz-Sobolev space setting

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Abstract

In this paper, we establish the existence of three weak solutions for quasilinear elliptic equations in an Orlicz-Sobolev space via an abstract result recently obtained by Ricceri in [13].

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References

  1. Adams, R., Fournier, J.F. Sobolev Spaces (Second Edition). Acad. Press, New York, 2003

    MATH  Google Scholar 

  2. Ambrosetti, A., Azorero, G.J., Peral, I. Multiplicity for some nonlinear elliptic equations. J. Funct. Anal., 137 (1): 219–242 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Clément, Ph., García-Huidobro, M., Manásevich, K. Schmitt, R. Moutain pass type solutions for quasilinear elliptic equations. Calc. Var. Partial Differential Equations, 11 (1): 33–62 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Damascelli, L., Sciunzi, B. Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Differential Equations, 206 (2): 483–515 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Damascelli, L., Sciunzi, B. Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations. Calc. Var. Partial Differential Equations, 25 (2): 139–159 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Degiovanni, M., Lancelotti, S. Linking over cones and nontrivial solutions for p–Laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré: Analyse Non Linéaire, 24 (6): 907–919 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fukagai, N., Ito, M., Narukawa, K. Positive solutions of quasilinear elliptic equations with critical Orlicz- Sobolev nonlinearity on RN.Funkcialaj Ekvacioj, 49 (2): 235–267 (2006)

  8. Fukagai, N., Narukawa, K. On the existence of Multiple positive solutions of quasilinear elliptic eigenvalue problems. Annadli di Matematica, 186 (3): 539–564 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. García-Huidobro, M., Le, V., Manásevich, R., Schmitt, K. On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting. Nonlinear Differential Equations and Appl., 6 (2): 207–225 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Q. Existence of three solutions for p(x)-Laplacian equations. Nonlinear Anal., 68 (7): 2119–2127 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rao, M.M., Ren, Z.D. Theory of Orlicz Spaces. Marcel Dekker, New York, 1985

    MATH  Google Scholar 

  12. Liu, S.B. Existence of solutions to a superlinear p-Laplacian equation. Electron. J. Differential Eq., 66: 1–6 (2001)

    MathSciNet  Google Scholar 

  13. Ricceri, B. A further three critical points theorem. Nonlinear Anal., 71 (9): 4151–4157 (2009)

  14. Ricceri, B. On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Optim., 46 (4): 543–549 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sun, J. T., Chen, H.B., Nieto, J.J., Otero-Novoa, M. The multiplicity of solutions for perturbed secondorder Hamiltonian systems with impulsive effects. Nonlinear Anal., 72 (12): 4575–4586 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for carefully reading the paper and for giving us valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Technology and Business University, Beijing, 100084, China

    Fei Fang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Zhong Tan

Authors
  1. Fei Fang
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  2. Zhong Tan
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Correspondence to Zhong Tan.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 11626038).

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Fang, F., Tan, Z. Existence of three solutions for quasilinear elliptic equations: an Orlicz-Sobolev space setting. Acta Math. Appl. Sin. Engl. Ser. 33, 287–296 (2017). https://doi.org/10.1007/s10255-017-0659-0

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  • DOI: https://doi.org/10.1007/s10255-017-0659-0

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