Publication Cover
Applicable Analysis
An International Journal
Volume 96, 2017 - Issue 13
277
Views
2
CrossRef citations to date
0
Altmetric
Articles

Semiclassical ground states for a class of nonlinear Kirchhoff-type problems

, &
Pages 2267-2284 | Received 06 Apr 2016, Accepted 27 Jul 2016, Published online: 16 Aug 2016

References

  • Kirchhoff G. Mechanik. Leipzig: Teubner; 1883.
  • Bernstein S. Sur une classe d’\’{e}quations fonctionelles aux dérivées partielles. Bull. Acad. Sci. URSS. Sér. Math. 1940;4:17–26.
  • Poho\u{z}aev SI. A certain class of quasilinear hyperbolic equations. Mat. Sb. (N.S.). 1975;96:152–166. Russian.
  • Lions JL. On some questions in boundary value problems of mathematical physics. In: de la Penha GM, Medeiros LA, editor. Contemporary development in continuum mechanics and partial differential equations. Vol. 30, North-Holland mathematics studies. Amsterdam: North-Holland; 1978. pp. 284–346.
  • Perera K, Zhang ZT. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 2006;221:246–255.
  • 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.
  • Liang ZP, Li FY, Shi JP. Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Ann. I. H. Poincaré-AN. 2014;31:155–167.
  • Li GB, Ye HY. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in ℝ3. J. Differ. Equ. 2014;257:566–600.
  • Liang SH, Zhang JH. Existence of solutions for Kirchhoff type problems with critical nonlinearity in ℝ3. Nonlinear Anal. 2014;17:126–136.
  • Nie JJ. Existence and multiplicity of nontrivial solutions for a class of Schrödinger-Kirchhoff-type equations. J. Math. Anal. Appl. 2014;417:65–79.
  • Sun JT, Wu TF. Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 2014;256:1771–1792.
  • Wu X. Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in ℝ3. Nonlinear. Anal. 2011;12:1278–1287.
  • Zhang J, Tang XH, Zhang W. Existence of multiple solutions of Kirchhoff type equation with sign-changing potential. Appl. Math. Comput. 2014;242:491–499.
  • Figueiredo GM, Ikoma N, Santos Júnior JR. Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Rat. Mech. Anal. 2014;213:931–979.
  • He XM, Zou WM. Existence and concentration behavior of positive solutions for a Kirchhoff equation in ℝ3. J. Differ. Equ. 2012;2:1813–1834.
  • He Y, Li GB. Standing waves for a class of Kirchhoff type problems in ℝ3 involving critical Sobolev exponents. Calc. Var. Partial Differ. Equ. 2015;54:3067–3106.
  • He Y, Li GB, Peng SJ. Concentrating bound states for Kirchhoff type problems in ℝ3 involving critical Sobolev exponents. Adv. Nonlinear Stud. 2014;14:483–510.
  • 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.
  • Ding YH, Liu XY. Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities. Manuscripta Math. 2013;140:51–82.
  • Rabinowitz P. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992;43:270–291.
  • Szulkin A, Weth T. The method of Nehari manifold. Handbook of nonconvex analysis and applications. Somerville (MA): Int. Press; 2010.
  • Huang YS, Wu TF, Wu YZ. Multiple positive solutions for a class of concave--convex elliptic problems in ℝN involving sign-changing weight, II. Commun. Contemp. Math. 2015;17:1450045. 35 p.
  • Sun JT, Wu TF, Feng ZS. Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system. J. Differ. Equ. 2016;260:586–627.
  • Wu YZ, Huang YS, Liu Z. On a Kirchhoff type problem in ℝN. J. Math. Anal. Appl. 2015;425:548–564.
  • Fang XD, Szulkin A. Multiple solutions for a quasilinear Schrödinger equation. J. Differ. Equ. 2013;254:2015–2032.
  • Zhang H, Zhang FB. Ground states for the nonlinear Kirchhoff type problems. J. Math. Anal. Appl. 2015;423:1671–1692.
  • He XM, Zou WM. Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J. Math. Phys. 2012;53:023702.
  • Pankov A. On decay of solution to nonlinear Schrödinger equations. Proc. Am. Math. Soc. 2008;136:2565–2570.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer-Verlag; 1983.

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.