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Solutions of the Quasilinear Elliptic Systems with Combined Critical Sobolev–Hardy Terms

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Ukrainian Mathematical Journal Aims and scope

We study the existence of multiple solutions for a quasilinear elliptic system. Based on the Ambrosetti–Rabinowitz mountain-pass theorem and the Rabinowitz symmetric mountain-pass theorem, we establish several existence and multiplicity results for the solutions and G-symmetric solutions under certain suitable conditions.

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References

  1. L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequality with weights,” Compos. Math., 53, 259–275 (1984).

    MathSciNet  MATH  Google Scholar 

  2. B. Xuan, “The solvability of quasilinear Brezis–Nirenberg-type problems with singular weights,” Nonlin. Anal., 62, 703–725 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Secchi, D. Smets, and M. Willem, “Remarks on a Hardy–Sobolev inequality,” Compt. Rend. Math., 336, 811–815 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Kang, “Positive solutions to the weighted critical quasilinear problems,” Appl. Math. Comput., 213, No. 2, 432–439 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Alves, D. Filho, and M. Souto, “On systems of elliptic equations involving subcritical or critical Sobolev exponents,” Nonlin. Anal., 42, 771–787 (2000).

    Article  MATH  Google Scholar 

  6. Z. Deng and Y. Huang, “Existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical Hardy–Sobolev exponents,” J. Math. Anal. Appl., 393, 273–284 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  7. N. Ghoussoub and F. Robert, “The effect of curvature on the best constant in the Hardy–Sobolev inequality,” Geom. Funct. Anal., 16, 897–908 (2006).

    Article  MathSciNet  Google Scholar 

  8. D. Kang and S. Peng, “Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential,” Appl. Math. Lett., 18, 1094–1100 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Han, “Multiple positive solutions for a critical growth problem with Hardy potential,” Proc. Edinburgh Math. Soc., 49, 53–69 (2006).

    Article  MATH  Google Scholar 

  10. D. Cao and D. Kang, “Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms,” J. Math. Anal. Appl., 333, 889–903 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Filippucci, P. Pucci, and F. Robert, “On a p-Laplace equation with multiple critical nonlinearities,” J. Math. Pures Appl., 91, 156–177 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Xuan and J. Wang, “Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents,” Nonlin. Anal., 72, 3649–3658 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  13. T. S. Hsu and H. L. Lin, “Multiplicity of positive solutions for weighted quasilinear elliptic equations involving critical Hardy–Sobolev exponents and concave-convex nonlinearities,” Abstract Appl. Anal., doi:10.1155/2012/579481 (2012).

  14. H. Rabinowitz, Methods in Critical Point Theory with applications to Differential Equations, CBMS, Amer. Math. Soc. (1986).

  15. P. L. Lions, “The concentration-compactness principle in the calculus of variations: the limit case. Pt 1,” Rev. Mat. Iberoam, 1, 145–201 (1985).

    Article  MATH  Google Scholar 

  16. J. Mawhin and M. Willem, “Critical point theory and Hamiltonian systems,” Appl. Math. Sci., 74 (1989).

  17. Y. Deng and L. Deng, “On symmetric solutions of a singular elliptic equation with critical Sobolev–Hardy exponent,” J. Math. Anal. Appl., 329, 603–616 (2007).

    Article  MathSciNet  MATH  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 6, pp. 788–808, June, 2015.

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Nyamoradi, N. Solutions of the Quasilinear Elliptic Systems with Combined Critical Sobolev–Hardy Terms. Ukr Math J 67, 891–915 (2015). https://doi.org/10.1007/s11253-015-1121-1

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  • DOI: https://doi.org/10.1007/s11253-015-1121-1

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