Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 7
302
Views
9
CrossRef citations to date
0
Altmetric
Articles

Nehari-type ground state solutions for Kirchhoff type problems in ℝN

, &
Pages 1255-1266 | Received 30 Oct 2017, Accepted 11 Dec 2017, Published online: 04 Jan 2018

References

  • Kirchhoff G . Mechanik. Leipzig: Teubner; 1883.
  • Chipot M , Lovat B . Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 1997;30(7):4619–4627.
  • Arosio A , Panizzi S . On the well-posedness of the Kirchhoff string. Trans Amer Math Soc. 1996;348:305–330.
  • Cavalcanti MM , Domingos Cavalcanti VN , Soriano JA . Global existence and uniform decay rates for the Kirchhoff--Carrier equation with nonlinear dissipation. Adv Differ Equ. 2001;6:701–730.
  • D’Ancona P , Spagnolo S . Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math. 1992;108:247–262.
  • Lions JL . 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, 1997. Vol. 30, North-Holland mathematics studies. Amsterdam: North-Holland; 1978. p. 284–346
  • Alves CO , Corrêa FJSA , Ma TF . Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput Math Appl. 2005;49:85–93.
  • Cheng BT , Wu X . Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal. 2009;71:4883–4892.
  • Chen CY , Kuo YC , Wu TF . The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J Differ Equ. 2011;250:1876–1908.
  • Figueiredo GM . Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J Math Anal Appl. 2013;401:706–713.
  • He XM , Zou WM . Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 2009;70:1407–1414.
  • Mao AM , Zhang ZT . Sign-changing and multiple solutions of Kirchhoff type problems without the P. S. condition. Nonlinear Anal. 2009;70(3):1275–1287.
  • Naimen D . The critical problem of Kirchhoff type elliptic equations in dimension four. J Differ Equ. 2014;257:1168–1193.
  • Perera K , Zhang ZT . Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Differ Equ. 2006;221:246–255.
  • Shuai W . Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J Differ Equ. 2015;259:1256–1274.
  • Tang XH . Non-Nehari manifold method for superlinear Schrödinger equation. Taiwanese J Math. 2014;18:1950-1972.
  • Xie Q , Wu XP , Tang CL . Existence of solutions for Kirchhoff type equations. Electron J Differ Equ. 2015;47:1–8.
  • Zhang ZT , Perera K . Sign changing solutions of Kirchhoff type problems via invarint sets of descent flow. J Math Anal Appl. 2006;317(2):456–463.
  • Jin J , Wu X . Infinitely many radial solutions for Kirchhoff-type problems in ℝ N . J Math Anal Appl. 2010;369:564–574.
  • Wu X . Existence of nontrivial solutions and high energy solutions for Schrödinger--Kirchhoff-type equations in ℝ 3 . Nonlinear Anal Real World Appl. 2011;12:1278–1287.
  • Li YH , Li FY , Shi JP . Existence of a positive solution to Kirchhoff type problems without compactness conditions. J Differ Equ. 2012;253:2285–2294.
  • He XM , Zou WM . Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ 3 . J Differ Equ. 2012;2:1813–1834.
  • Li GB , Ye HY . Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ 3 . J Differ Equ. 2014;257:566–600.
  • Jeanjean L . On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type probleme set on ℝ N . Proc Roy Soc Edinburgh Sect A. 1999;129:787–809.
  • Wang J , Tian LX , Xu JX , et al . Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J Differ Equ. 2012;253:2314–2351.
  • He Y , Li GB , Peng SJ . Concentrating bound states for Kirchhoff type problems in ℝ 3 involving critical Sobolev exponents. Adv Nonlinear Stud. 2014;14:483–510.
  • He Y , Li GB . Standing waves for a class of Kirchhoff type problems in ℝ 3 involving critical Sobolev exponents. Calc Var Partial Differ Equ. 2015;54(3):3067–3106.
  • Figueiredo GM , Ikoma N , Santos JR Jr . Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch Rat Mech Anal. 2014;213:931–979.
  • Ruiz D . The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal. 2006;237:655–674.
  • Guo Z . Ground states for Kirchhoff equations without compact condition. J Differ Equ. 2015;259:2884–2902.
  • Alves CO , Figueiredo GM . Nonlinear perturbations of a periodic Kirchhoff equation in ℝ N . Nonlinear Anal. 2012;75:2750–2759.
  • Ma XY , He XM . Nontrivial solutions for Kirchhoff equations with periodic potentials. Electron J Differ Equ. 2016;102:1–22.
  • Xie Q , Ma S . Existence and concentration of positive solutions for Kirchhoff-type problems with a steep well potential. J Math Anal Appl. 2015;431:1210–1223.
  • Tang XH , Cheng BT . Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J Differ Equ. 2016;261:2384–2402.
  • Lions PL . The concentration-compactness principle in the calculus of variations. The locally compact case. I & II. Ann Inst H Poincaré Anal Non Linéaire. 1984;1:109–145, 223–283.
  • Willem M . Minimax theorems. Boston: Birkhäuser; 1996.

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:

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:

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.