Taiwanese Journal of Mathematics

Ground State Solutions for Kirchhoff-type Problems with Critical Nonlinearity

Yiwei Ye

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

In this paper, we study the Kirchhoff-type equation with critical exponent \[ -\left( a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx \right) \Delta u + V(x)u = a(x) f(u) + u^5 \quad \textrm{in $\mathbb{R}^3$}, \] where $a,b \gt 0$ are constants, $V \in C(\mathbb{R}^3,\mathbb{R})$, $\lim_{|x| \to \infty} V(x) = V_{\infty} \gt 0$ and $V(x) \leq V_{\infty} + C_1 e^{-b |x|}$ for some $C_1 \gt 0$ and $|x|$ large enough. Via variational methods, we prove the existence of ground state solution.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 17 pages.

Dates
First available in Project Euclid: 15 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1555315213

Digital Object Identifier
doi:10.11650/tjm/190402

Subjects
Primary: 45M20: Positive solutions
Secondary: 35B33: Critical exponents

Keywords
Kirchhoff-type equation ground state solutions variational methods

Citation

Ye, Yiwei. Ground State Solutions for Kirchhoff-type Problems with Critical Nonlinearity. Taiwanese J. Math., advance publication, 15 April 2019. doi:10.11650/tjm/190402. https://projecteuclid.org/euclid.twjm/1555315213


Export citation

References

  • C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in $\mathbb{R}^N$, Nonlinear Anal. 75 (2012), no. 5, 2750–2759.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381.
  • J. Chen, Multiple positive solutions to a class of Kirchhoff equation on $\mathbb{R}^3$ with indefinite nonlinearity, Nonlinear Anal. 96 (2014), 134–145.
  • 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.
  • Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Adv. Nonlinear Stud. 14 (2014), no. 2, 483–510.
  • 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.
  • G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
  • G. Li and H.-S. Zhou, The existence of a positive solution to asymptotically linear scalar field equations, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 1, 81–105.
  • L. Li and J.-J. Sun, Existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities, Nonlinear Anal. Real World Appl. 26 (2015), 391–399.
  • J.-L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), 284–346, North-Holland Math. Stud. 30, North-Holland, Amsterdam, 1978.
  • P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97.
  • Z. Liu and S. Guo, On ground states for the Kirchhoff-type problem with a general critical nonlinearity, J. Math. Anal. Appl. 426 (2015), no. 1, 267–287.
  • 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.
  • D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differential Equations Appl. 21 (2014), no. 6, 885–914.
  • ––––, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 (2014), no. 4, 1168–1193.
  • S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96(138) (1975), 152–166.
  • 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.
  • M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications 24, Birkhäuser Boston, Boston, MA, 1996.
  • Q. Xie, Singular perturbed Kirchhoff type problem with critical exponent, J. Math. Anal. Appl. 454 (2017), no. 1, 144–180.
  • Q.-L. Xie, X.-P. Wu and C.-L. Tang, Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2773–2786.
  • L. Yang, Z. Liu and Z. Ouyang, Multiplicity results for the Kirchhoff type equations with critical growth, Appl. Math. Lett. 63 (2017), 118–123.
  • Y. Ye and C.-L. Tang, Multiple solutions for Kirchhoff-type equations in $\mathbb{R}^N$, J. Math. Phys. 54 (2013), no. 8, 081508, 16 pp.
  • J. Zhang, On the Schrödinger equations with a nonlinearity in the critical growth, Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 457–469.
  • ––––, The Kirchhoff type Schrödinger problem with critical growth, Nonlinear Anal. Real World Appl. 28 (2016), 153–170.
  • J. Zhang and W. Zou, Multiplicity and concentration behavior of solutions to the critical Kirchhoff-type problem, Z. Angew. Math. Phys. 68 (2017), no. 3, Art. 57, 27 pp.
  • X.-J. Zhong and C.-L. Tang, Multiple positive solutions to a Kirchhoff type problem involving a critical nonlinearity, Comput. Math. Appl. 72 (2016), no. 12, 2865–2877.
  • X. P. Zhu and D. M. Cao, The concentration-compactness principle in nonlinear elliptic equations, Acta Math. Sci. (English Ed.) 9 (1989), no. 3, 307–328.