Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

POSITIVE SOLUTION AND GROUND STATE SOLUTION FOR A KIRCHHOFF TYPE EQUATION WITH CRITICAL GROWTH

  • Chen, Caixia (Mathematics Department of Jining University) ;
  • Qian, Aixia (School of Mathematical Sciences Qufu Normal University)
  • Received : 2021.08.02
  • Accepted : 2021.11.08
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider the following Kirchhoff type equation on the whole space $$\{-(a+b{\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}\;{\mid}{\nabla}u{\mid}^2dx){\Delta}u=u^5+{\lambda}k(x)g(u),\;x{\in}{\mathbb{R}}^3,\\u{\in}{\mathcal{D}}^{1,2}({\mathbb{R}}^3),$$ where λ > 0 is a real number and k, g satisfy some conditions. We mainly investigate the existence of ground state solution via variational method and concentration-compactness principle.

Keywords

Acknowledgement

The authors would like to express sincere thanks to the anonymous referee for his/her carefully reading of the manuscript and valuable comments and suggestions.

References

  1. G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrodinger-Poisson systems, J. Differential Equations 248 (2010), no. 3, 521-543. https://doi.org/10.1016/j.jde.2009.06.017
  2. 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), no. 2, 706-713. https://doi.org/10.1016/j.jmaa.2012.12.053
  3. X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ3, J. Differential Equations 252 (2012), no. 2, 1813-1834. https://doi.org/10.1016/j.jde.2011.08.035
  4. T. Hu and L. Lu, Multiplicity of positive solutions for Kirchhoff type problems in ℝ3, Topol. Methods Nonlinear Anal. 50 (2017), no. 1, 231-252. https://doi.org/10.12775/tmna.2017.028
  5. Y. Huang, Z. Liu, and Y. Wu, On finding solutions of a Kirchhoff type problem, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3019-3033. https://doi.org/10.1090/proc/12946
  6. L. Huang, E. M. Rocha, and J. Chen, On the Schrodinger-Poisson system with a general indefinite nonlinearity, Nonlinear Anal. Real World Appl. 28 (2016), 1-19. https://doi.org/10.1016/j.nonrwa.2015.09.001
  7. C.-Y. Lei, J.-F. Liao, and C.-L. Tang, Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents, J. Math. Anal. Appl. 421 (2015), no. 1, 521-538. https://doi.org/10.1016/j.jmaa.2014.07.031
  8. C.-Y. Lei, G.-S. Liu, and L.-T. Guo, Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal. Real World Appl. 31 (2016), 343-355. https://doi.org/10.1016/j.nonrwa.2016.01.018
  9. C.-Y. Lei, H. Suo, C. Chu, and L. Guo, On ground state solutions for a Kirchhoff type equation with critical growth, Comput. Math. Appl. 72 (2016), no. 3, 729-740. https://doi.org/10.1016/j.camwa.2016.05.027
  10. G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ3, J. Differential Equations 257 (2014), no. 2, 566-600. https://doi.org/10.1016/j.jde.2014.04.011
  11. S. Liang and J. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in ℝ3, Nonlinear Anal. Real World Appl. 17 (2014), 126-136. https://doi.org/10.1016/j.nonrwa.2013.10.011
  12. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201. https://doi.org/10.4171/RMI/6
  13. P.-L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Part II, Rev. Mat. Iberoam. 2 (1985), no. 2, 45-121. https://doi.org/10.4171/RMI/12
  14. Z. Liu and C. Luo, Existence of positive ground state solutions for Kirchhoff type equation with general critical growth, Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 165-182. https://doi.org/10.12775/tmna.2016.068
  15. Z. Liu, Z.-Q. Wang, and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrodinger-Poisson system, Ann. Mat. Pura Appl. (4) 195 (2016), no. 3, 775-794. https://doi.org/10.1007/s10231-015-0489-8
  16. A. Ourraoui, On a p-Kirchhoff problem involving a critical nonlinearity, C. R. Math. Acad. Sci. Paris 352 (2014), no. 4, 295-298. https://doi.org/10.1016/j.crma.2014.01.015
  17. A. Qian, J. Liu, and A. Mao, Ground state and nodal solutions for a Schrodinger-Poisson equation with critical growth, J. Math. Phys. 59 (2018), no. 12, 121509, 20 pp. https://doi.org/10.1063/1.5050856
  18. M. Shao and A. Mao, Signed and sign-changing solutions of Kirchhoff type problems, J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 2, 20 pp. https://doi.org/10.1007/s11784-018-0486-9
  19. J. Wang, L. Tian, J. Xu, and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), no. 7, 2314-2351. https://doi.org/10.1016/j.jde.2012.05.023
  20. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhauser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1