Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

ON KIRCHHOFF TYPE EQUATIONS WITH SINGULAR NONLINEARITIES, SUB-CRITICAL AND CRITICAL EXPONENT

  • Received : 2023.06.07
  • Accepted : 2023.10.02
  • Published : 2024.06.15
Download PDF
⟨ Previous Next ⟩

Abstract

This paper is devoted to the existence of solutions for Kirchhoff type equations with singular nonlinearities, sub-critical and critical exponent. By using the Nehari manifold and Maximum principle theorem, the existence of at least two distinct positive solutions is obtained.

Keywords

Acknowledgement

The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number(4524) during the academic year 1445AH/2024AD.

References

  1. Z. I. Almuhiameed, Existence results for p-Laplacian problems involving singular cylindrical potential , Nonlinear Funct. Anal. Appl., 28(4) (2023), 1005-1015.
  2. C. Alves, F. Correa and T. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. https://doi.org/10.1016/j.camwa.2005.01.008
  3. K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign changing weight function, J. Diff. Equ., 2 (2003), 481-499. https://doi.org/10.1016/S0022-0396(03)00121-9
  4. B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl., 394 (2012), 488-495. https://doi.org/10.1016/j.jmaa.2012.04.025
  5. B. Cheng and X. Wu, Existence results of positive solutions of Krichhoff problems, Nonlinear Anal., 71 (2009), 4883-4892. https://doi.org/10.1016/j.na.2009.03.065
  6. M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30 (1997), 4619-4627. https://doi.org/10.1016/S0362-546X(97)00169-7
  7. F.J S.A. Correa, S.D.B. Menezes and J. Ferreira, On a class of problems involving a nonlocal operator, Appl. Math. Comput., 147 (2004), 475-489. https://doi.org/10.1016/S0096-3003(02)00740-3
  8. P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. https://doi.org/10.1007/BF02100605
  9. M.E.O. El Mokhtar and A. Matallah, Existence of Multiple Positive Solutions for Brezis-Nirenberg-Type Problems Involving Singular Nonlinearities, J. Math., 2021 (2021) 1-8.
  10. M. Haddaoui, N. Tsouli and A. Zaki, Study of a critical 𝚽-Kirchhoff type equations in Orlicz-Sobolev spaces, Nonlinear Funct. Anal. Appl., 27(3) (2022), 641-648.
  11. D. Kang and S. Peng, Positive solutions for singular elliptic problems, Appl. Math. Lett., 17 (2004), 411-416. https://doi.org/10.1016/S0893-9659(04)90082-1
  12. C. Lei, J. Liao and C. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl., 421 (2015), 521-538. https://doi.org/10.1016/j.jmaa.2014.07.031
  13. Y. Li, F. Li and J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass, J. Math. Anal. Appl., 410 (2014), 361-374. https://doi.org/10.1016/j.jmaa.2013.08.030
  14. J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations in: North-HollandMath. Stud. North-Holland. Amsterdam, 30 (1978), 284-346. https://doi.org/10.1016/S0304-0208(08)70870-3
  15. X. Liu and Y. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal., 12 (2013), 721-733. https://doi.org/10.3934/cpaa.2013.12.721
  16. A. Mao and S. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. https://doi.org/10.1016/j.jmaa.2011.05.021
  17. A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. https://doi.org/10.1016/j.na.2008.02.011
  18. K. Sabri, M. El Mokhtar Ould El Mokhtar and A. Matallah, Multiple nontrivial solutions for critical p-Kirchhoff type problems in RN, Nonlinear Funct. Anal. Appl., 29(1) (2024), 35-45.
  19. J. Simon, Sur des equations aux derivees partielles nonlineaires, These, Paris, 1977.
  20. J. Sun and C. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74 (2011), 1212-1222. https://doi.org/10.1016/j.na.2010.09.061
  21. S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Diff. Equ., 1 (1996), 241-264.
  22. Q. Xie, X. Wu and C. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal., 12 (2013), 2773-2786 https://doi.org/10.3934/cpaa.2013.12.2773
  23. Z. Zhang and K. Perera, Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317 (2006), 456-463. https://doi.org/10.1016/j.jmaa.2005.06.102