Multiplicity result to some Kirchhoff-type biharmonic equation involving exponential growth conditions

Document Type: Research Paper


Institut Superieur des Mathematiques Appliquees et de l'Informa-tique de Kairouan‎, ‎3100 Kairouan‎, ‎Tunisia.


In this paper‎, ‎we prove a multiplicity result for some biharmonic elliptic equation of Kirchhoff type and involving nonlinearities with critical exponential growth at infinity‎. ‎Using some variational arguments and exploiting the symmetries of the problem‎, ‎we establish a multiplicity result giving two nontrivial solutions‎.


Main Subjects

S. Aouaoui, A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in R2; Commun. Pure Appl. Anal. 15 (2016), no. 4, 1351--1370.

S. Aouaoui, Existence of multiple solutions to elliptic problems of Kirchhoff type with critical exponential growth, Electon. J. Differential Equations 2014 (2014), no. 107, 12 pages.

S. Aouaoui, On some nonlocal problem involving the N-Laplacian in RN; Nonlinear Stud. 22 (2015), no. 1, 57--70.

G. Autuori, F. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Var. Elliptic Equ. 57 (2012), no. 2-4, 379--395.

T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal. 117 (1993), no. 2, 447--460.

I. Ekeland, On the variational principle, J. Math. Anal. App. 47 (1974) 324--353.

A. Ferrero and G. Warnault, On solutions of second and fourth order elliptic equations with power-type nonlinearities, Nonlinear Anal. 70 (2009), no. 8, 2889--2902.

S. Goyal, P. K. Mishra and K. Sreenadh, n-Kirchhoff type equations with exponential nonlinearities, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 110 (2016), no. 1, 219--245.

E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl. (9) 76 (1997), no. 10, 859--881.

J. Koboyashi and M. Otani, The principle of symmetric criticality for non-differentiable mappings, J. Funct. Anal. 214 (2004), no. 2, 428--449.

C. Li and C. L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. 72 (2010), no. 3-4, 1339--1347.

Q. Li and Z. Yang, Multiple solutions for N-Kirchhoff type problems with critical exponential growth in RN; Nonlinear Anal. 117 (2015), no. 1, 159--168.

P. L. Lions, Sym_etrie et compacitfie dans les espaces de Sobolev, J. Funct. Anal. 49

(1982), no. 3, 315--334.

T. G. Myers, Thin films with high surface tension, SIAM Rev. 40 (1998), no. 3, 441--462.

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19--30.

B. Ruf and F. Sani, Sharp Adams-type inequalities in RN; Trans. Amer. Math. Soc. 365 (2013), no. 2, 645--670.

F. Sani, A biharmonic equation in R4 involving nonlinearities with critical exponential growth, Commun. Pure Appl. Anal. 12 (2013), no. 1, 405--428.

F. Sani, A biharmonic equation in R4 involving nonlinearities with subcritical exponential growth, Adv. Nonlinear Stud. 11 (2011), no. 4, 889--904.

N. S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967) 721--747.

F. Wang, T. An and Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type on RN; Electron. J. Qual. Theory Differ. Equ. 39 (2014) 11 pages.

F. Wang, M. Avci and Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J. Math. Anal. Appl. 409 (2014), no. 1, 140--146.

W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl. 348 (2008), no. 2, 730--738.

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhauser, Boston, 1996.

L. Xu and H. Chen, Existence and multiplicity of solutions for fourth-order ellptic equations of Kirchhoff type via genus theory, Bound. Value Probl. 2014 (2014), 12 pages.

Volume 42, Issue 6
November and December 2016
Pages 1559-1569
  • Receive Date: 11 May 2015
  • Revise Date: 29 September 2015
  • Accept Date: 29 September 2015