Advertisement

Results in Mathematics

, 74:42 | Cite as

Groundstates for Kirchhoff-Type Equations with Hartree-Type Nonlinearities

  • Yan Li
  • Xinfu Li
  • Shiwang MaEmail author
Article
  • 82 Downloads

Abstract

In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities:
$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^N}|Du|^2\right) \Delta u+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,&{}\quad x\in \mathbb {R}^N,\\ \\ u\in H^1(\mathbb {R}^N),\quad u>0,&{}\quad x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$
where \(N\ge 3\), \(\max \{0,N-4\}<\alpha <N\), \(2<p<\frac{N+\alpha }{N-2}\), \(a>0,b\ge 0\) are constants, \(I_{\alpha }\) is the Riesz potential and \(V{:}\,\mathbb {R}^N\rightarrow \mathbb {R}\) is a potential function. Under certain assumptions on V, we prove that the problem has a positive ground state solution by using global compactness lemma, monotonicity technique and some new tricks recently given in the literature.

Keywords

Kirchhoff equations positive solution Pohožaev identity ground state solution Hartree-type nonlinearity 

Mathematics Subject Classification

35J20 35J60 

Notes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 11571187) and Tianjin Municipal Education Commission with the Grant No. 2017KJ173 “Qualitative studies of solutions for two kinds of nonlocal elliptic equations”.

References

  1. 1.
    Guo, Z.J.: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884–2902 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    He, X.M., Zou, W.M.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^3\). J. Differ. Equ. 252, 1813–1834 (2012)CrossRefGoogle Scholar
  3. 3.
    Jin, J.H., Wu, X.: Infinitely many radial solutions for Kirchhoff-type problems in \(\mathbb{R}^N\). J. Math. Anal. Appl. 369, 564–574 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jeanjean, L.: On the existence of bounded Palais–Smale sequences and application to a Landsman–Lazer-type problem set on \(\mathbb{R}^N\). Proc. Edinb. Math. Soc. 2(129), 787–809 (1999)CrossRefGoogle Scholar
  5. 5.
    Lieb, E.H., Loss, M.: Analysis, Volume 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, (4) (2001)Google Scholar
  6. 6.
    Li, G., Ye, H.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^3\). J. Differ. Equ. 257, 566–600 (2014)CrossRefGoogle Scholar
  7. 7.
    Li, X., Ma, S., Zhang, G.: Existence and qualitative properties of solutions for the Choquard equations with a local term. Nonlinear Anal. Real World Appl. 45, 1–25 (2019)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, W., He, X.M.: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, 473–487 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lü, D.F.: A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 99, 35–48 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 256, 153–184 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tang, X.H., Chen, S.: Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials. Calc. Var. 56(110), 1–25 (2017)zbMATHGoogle Scholar
  13. 13.
    Wang, J., et al.: Multiplicity and concentration of positive solutions for Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)CrossRefGoogle Scholar
  15. 15.
    Willem, M.: Functional Analysis: Fundamentals and Applications (Cornerstones), vol. XIV. Birkhäuser, Basel (2013)CrossRefGoogle Scholar
  16. 16.
    Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^N\). Nonlinear Anal. Real World Appl. 12, 1278–1287 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina
  2. 2.School of ScienceTianjin University of CommerceTianjinChina

Personalised recommendations