Nonlinear Functional Analysis and Applications

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

DOI QR Code

MULTIPLE NONTRIVIAL SOLUTIONS FOR CRITICAL p-KIRCHHOFF TYPE PROBLEMS IN ℝN

  • Khadidja Sabri (University of Oran 2, Laboratory of Analysis and Control of Partial Differential Equations) ;
  • Mohammed El Mokhtar Ould El Mokhtar (Department of Mathematics, College of Science, Qassim University) ;
  • Atika Matallah (Higher School of Management-Tlemcen, Laboratory of Analysis and Control of Partial Differential Equations)
  • Received : 2022.10.22
  • Accepted : 2023.02.28
  • Published : 2024.03.15
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study the existence and multiplicity of nontrivial solutions for a p-Kirchhoff equation involving critical Sobolev-Hardy exponent by using variational methods and we need to estimate the energy levels.

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 (1125) during the academic year 1443AH/2021AD. And also, Algerian Ministry of Higher Education and Scientific Research on the material support for this research under the number (1125) during the academic year 1443AH/2021AD.

References

  1. C.O. Alves, Multiple positive solutions for equations involving critical Sobolev exponent, Electron. J. Diff. Equ., 13 (1997), 1-10.
  2. C.O. Alves, F.J.S.A. Correa and T.F. 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. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  4. A. Benaissa and A. Matallah, Nonhomogeneous elliptic Kirchhoff equations of the p-Laplacian Type, Ukr. Math. Math. J., 72 (2020), 203-210. https://doi.org/10.1007/s11253-020-01776-z
  5. H. Brezis and 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
  6. L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53(3) (1984) 259-275.
  7. C.S. Chen, J.C. Huang and L.H. Liu, Multiple solutions to the nonhomogeneous p-Kirchhoff elliptic equation with concave-convex nonlinearities, Appl. Math. Lett., 26 (2013), 754-759. https://doi.org/10.1016/j.aml.2013.02.011
  8. S.J. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on ℝN, Nonlinear Anal. Real World Appl., 14 (2013), 1477-1486 https://doi.org/10.1016/j.nonrwa.2012.10.010
  9. F.J.S.A. Correa and G.M. Figueiredo, On a elliptic equation of p-Kirchhoff type via variational methods, Bull. Aust. Math. Soc., 74 (2006), 263-277. https://doi.org/10.1017/S000497270003570X
  10. I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0
  11. G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706-713. https://doi.org/10.1016/j.jmaa.2012.12.053
  12. A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. https://doi.org/10.1016/j.na.2013.08.011
  13. N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. https://doi.org/10.1090/S0002-9947-00-02560-5
  14. X. Wu, Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in ℝN, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. https://doi.org/10.1016/j.nonrwa.2010.09.023