A p-Laplacian Elliptic System with Strongly Coupled Critical Terms and Concave-Convex Nonlinearities

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Mohamed Hatimi, Rachid Echarghaoui

Abstract

The main purpose of this paper is to establish some results on positive solutions for a p-Laplacian elliptic system with strongly coupled critical terms and concave nonlinearities. With the technique of variational method, namely Nehari manifold and Palais-Smale condition we show that there are at least two nontrivial solutions for our problem.

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References

  1. S. Benmouloud, R. Echarghaoui, Si.M. Sbaï, Multiplicity of Positive Solutions for a Critical Quasilinear Elliptic System with Concave and Convex Nonlinearities, J. Math. Anal. Appl. 396 (2012), 375–385. https://doi.org/10.1016/j.jmaa.2012.05.078.
  2. L. Brasco, S. Mosconi, M. Squassina, Optimal Decay of Extremals for the Fractional Sobolev Inequality, Calc. Var. Partial Differ. Equ. 55 (2016), 23. https://doi.org/10.1007/s00526-016-0958-y.
  3. H. Brézis, E. Lieb, A Relation between Pointwise Convergence of Functions and Convergence of Functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490. https://doi.org/10.1090/s0002-9939-1983-0699419-3.
  4. K. Benlhachmi, R. Echarghaoui, A. Hatimi, H. Hadad, Multiplicity of Positive Solutions for a Fractional Elliptic System With Strongly Coupled Critical Terms, Int. J. Anal. Appl. 22 (2024), 164. https://doi.org/10.28924/2291-8639-22-2024-164.
  5. R. Echarghaoui, A. Hatimi, M. Hatimi, A Fractional Elliptic System With Strongly Coupled Critical Terms and Concave-Convex Nonlinearities, Int. J. Anal. Appl. 22 (2024), 107. https://doi.org/10.28924/2291-8639-22-2024-107.
  6. P. Han, The Effect of the Domain Topology on the Number of Positive Solutions of Elliptic Systems Involving Critical Sobolev Exponents, Houston J. Math. 32 (2006), 1241-1257.
  7. T.-S. Hsu, Existence and Multiplicity of Positive Solutions to a Perturbed Singular Elliptic System Deriving from a Strongly Coupled Critical Potential, Bound. Value Probl. 2012 (2012), 116. https://doi.org/10.1186/1687-2770-2012-116.
  8. T.-S. Hsu, Multiple Positive Solutions for a Critical Quasilinear Elliptic System with Concave–Convex Nonlinearities, Nonlinear Anal.: Theory Methods Appl. 71 (2009), 2688–2698. https://doi.org/10.1016/j.na.2009.01.110.
  9. T.-S. Hsu, Multiplicity Results for p -Laplacian with Critical Nonlinearity of Concave-Convex Type and SignChanging Weight Functions, Abstr. Appl. Anal. 2009 (2009), 652109. https://doi.org/10.1155/2009/652109.
  10. Y. Huang, D. Kang, On the Singular Elliptic Systems Involving Multiple Critical Sobolev Exponents, Nonlinear Anal.: Theory Methods Appl. 74 (2011), 400–412. https://doi.org/10.1016/j.na.2010.08.051.
  11. D. Kang, M. Liu, L. Xu, Critical Elliptic Systems Involving Multiple Strongly–Coupled Hardy–Type Terms, Adv. Nonlinear Anal. 9 (2019), 866–881. https://doi.org/10.1515/anona-2020-0029.
  12. D. Kang, X. Liu, Singularities of Solutions to Elliptic Systems Involving Different Hardy-Type Terms, J. Math. Anal. Appl. 468 (2018), 757–765. https://doi.org/10.1016/j.jmaa.2018.08.044.
  13. G. Tarantello, On Nonhomogeneous Elliptic Equations Involving Critical Sobolev Exponent, Ann. Inst. H. Poincaré C Anal. Non Linéaire 9 (1992), 281–304. https://doi.org/10.1016/s0294-1449(16)30238-4.
  14. X. Zhou, H.-Y. Li, J.-F. Liao, Multiplicity of Positive Solutions for a Semilinear Elliptic System with Strongly Coupled Critical Terms and Concave Nonlinearities, Qual. Theory Dyn. Syst. 22 (2023), 126. https://doi.org/10.1007/s12346-023-00825-9.