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A degenerate elliptic system with variable exponents

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Abstract

We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial weak solutions of the system. Several consequences of the main theorem are derived; in particular, the existence of at lease two distinct nontrivial non-negative solutions is established for a scalar degenerate problem. One example is provided to show the applicability of our results.

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References

  1. Antontsev S N, Shmarev S I. A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions. Nonlinear Anal, 2005, 60: 515–545

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66: 1383–1406

    Article  MathSciNet  MATH  Google Scholar 

  3. Dancer E N, Du Y. Effects of certain degeneracies in the predator-prey model. SIAM J Math Anal, 2002, 34: 292–314

    Article  MathSciNet  MATH  Google Scholar 

  4. Dautray R, Lions J L. Mathematical Analysis and Numerical Methods for Science and Technology. Volume 1. Physical Origins and Classical Methods. Berlin: Springer-Verlag, 1985

    MATH  Google Scholar 

  5. Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Heidelberg: Springer, 2011

    Google Scholar 

  6. Drábek P, Kufner A, Nicolosi F. Quasilinear Elliptic Equations with Degenerations and Singularities. Berlin: Walter de Gruyter, 1997

    Book  MATH  Google Scholar 

  7. Fan X, Han X. Existence and multiplicity of solutions for p(x)-Laplacian equations in ℝN. Nonlinear Anal, 2004, 59: 173–188

    MathSciNet  MATH  Google Scholar 

  8. Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer, 1998

    MATH  Google Scholar 

  9. Ho K, Sim I. Existence and some properties of solutions for degenerate elliptic equations with exponent variable. Nonlinear Anal, 2014, 98: 146–164

    Article  MathSciNet  MATH  Google Scholar 

  10. Ho K, Sim I. Existence and multiplicity of solutions for degenerate p(x)-Laplace equations involving concave-convex type nonlinearities with two parameters. Taiwanese J Math, 2015, 19: 1469–1493

    Article  MathSciNet  MATH  Google Scholar 

  11. Ho K, Sim I. Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators. Sci China Math, 2017, 60: 133–146

    Article  MathSciNet  MATH  Google Scholar 

  12. Jabri Y. The Mountain Pass Theorem: Variants, Generalizations and Some Applications. Encyclopedia of Mathematics and Its Applications, vol. 95. New York: Cambridge University Press, 2003

    Google Scholar 

  13. Kim Y H, Wang L, Zhang C. Global bifurcation for a class of degenerate elliptic equations with variable exponents. J Math Anal Appl, 2010, 371: 624–637

    Article  MathSciNet  MATH  Google Scholar 

  14. Le V K. On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces. Nonlinear Anal, 2009, 71: 3305–3321

    Article  MathSciNet  MATH  Google Scholar 

  15. Mihăilescu M, Rădulescu V. A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proc R Soc Lond Ser A Math Phys Eng Sci, 2006, 462: 2625–2641

    Article  MathSciNet  MATH  Google Scholar 

  16. Rădulescu V, Repovš D. Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal, 2012, 75: 1524–1530

    Article  MathSciNet  MATH  Google Scholar 

  17. Råužička M. Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Berlin: Springer-Verlag, 2000

    Google Scholar 

  18. Zeidler Z. Nonlinear Functional Analysis and Its Applications. Volume III. New York: Springer, 1985

    Book  MATH  Google Scholar 

  19. Zhikov V. Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv, 1987, 29: 33–66

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by a University of Tennessee at Chattanooga SimCenter-Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant.

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  1. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA

    Lingju Kong

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  1. Lingju Kong
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Correspondence to Lingju Kong.

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Kong, L. A degenerate elliptic system with variable exponents. Sci. China Math. 62, 1373–1390 (2019). https://doi.org/10.1007/s11425-018-9409-5

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  • DOI: https://doi.org/10.1007/s11425-018-9409-5

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