Advanced search
Publication Cover
Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 12
Submit an article Journal homepage
276
Views
3
CrossRef citations to date
0
Altmetric
Articles

On positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ3

Zupei ShenCenter for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of ChinaView further author information
&
Jianshe YuCenter for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of ChinaCorrespondence[email protected]
View further author information
Pages 2137-2149 | Received 12 Mar 2018, Accepted 29 Nov 2018, Published online: 10 Dec 2018

References

  • Kirchhoff G. Mechanik, Teubner, Leipzig, 1883.
  • Lions JL. On some questions in boundary value problems of mathematical physics, in: Contemporary Development in Continuum Mechanics and Partial Differential Equations. in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York; 1978. p. 284–346.
  • Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans. Amer. Math. Soc. 1996;348(1):305–331. doi: 10.1090/S0002-9947-96-01532-2
  • D'Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 1992;108(2):247–262. doi: 10.1007/BF02100605
  • Deng Y, Peng S, Shuai W. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3. J. Funct. Anal. 2015;269:3500–3527. doi: 10.1016/j.jfa.2015.09.012
  • Figueiredo GM, Ikoma N, Santos, Jr JR Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities. Arch. Rational Mech. Anal. 2014;213:931–979. doi: 10.1007/s00205-014-0747-8
  • Figueiredo GM, Nascimento RG. Existence of a nodal solution with minimal energy for a Kirchho? equation. Mathematische Nachrichten. 2015;288:48–60. doi: 10.1002/mana.201300195
  • Li G, Ye H. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3. J. Differ. Equat. 2014;257(2):566–600. doi: 10.1016/j.jde.2014.04.011
  • Wang J, Tian L-X, Xu J-X, Zhang F-B. Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equat. 2012;253:2314–2351. doi: 10.1016/j.jde.2012.05.023
  • He XM, Zou WM. Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3. J. Differ. Equat. 2012;252:1813–1834. doi: 10.1016/j.jde.2011.08.035
  • Wang J, Xiao L. Existence and concentration of solutions for a kirchhoff type problem with potentials. Discrete Cont. Dynam. Syst. 2016;36:7137–7168. doi: 10.3934/dcds.2016111
  • Wu X. Existence of nontrivial solutions and high energy solutions for Schr&quote;odinger-Kirchhoff-type equations in RN. Nonlin. Anal. Real World Appl. 2011;12:1278–1287. doi: 10.1016/j.nonrwa.2010.09.023
  • Ye H. Positive high energy solution for Kirchhoff equation in R3 with superlinear nonlinearities via Nehari–Pohozaev manifold. Discrete Contin. Dyn. Syst. 2015;35(8):3857–3877. doi: 10.3934/dcds.2015.35.3857
  • Li Y, Ni WM. Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Comm. Partial Differ. Equat. 1993;18:1043–1054. doi: 10.1080/03605309308820960
  • Zhang H, Xu J, Zhang F. Semiclassical ground states for a class of nonlinear Kirchhoff-type problems. Appl. Anal. 2016;96(13):2267–2284. doi: 10.1080/00036811.2016.1220546
  • Lieb E, Loss M. Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc., Providence, RI, 2001.
  • Ruiz D. The Schro¨dinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 2006;237(2):655–674. doi: 10.1016/j.jfa.2006.04.005
  • Lehrer R, Maia LA. Positive solutions of asymptotically linear equations via Pohozaev manifold. J. Funct. Anal. 2014;266(1):213–246. doi: 10.1016/j.jfa.2013.09.002
  • Jeanjean L, Tanaka K. A remark on least energy solutions in RN. Proc. Amer. Math. Soc.; 2003;131(8):2399–2408. doi: 10.1090/S0002-9939-02-06821-1
  • Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973;14:349–381. doi: 10.1016/0022-1236(73)90051-7

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.