Acta mathematica scientia, Series B >
MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE
Received date: 2014-01-17
Revised date: 2015-05-07
Online published: 2015-09-01
Supported by
This work was supported by Natural Science Foundation of China (11271372) and Hunan Provincial Natural Science Foundation of China (12JJ2004).
In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type
???11???
where a,b>0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti-Rabinowitz type condition.
Liping XU , Haibo CHEN . MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE[J]. Acta mathematica scientia, Series B, 2015 , 35(5) : 1067 -1076 . DOI: 10.1016/S0252-9602(15)30040-0
[1] Ball J M. Initial-boundary value for an extensible beam. J Math Anal Appl, 1973, 42:61-90
[2] Berger H M. A new approach to the analysis of large deflections of plates. J Appl Mech, 1955, 22:465-472
[3] Ma T F. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Discrete Contin Dyn Syst, 2007, (Supplement):694-703
[4] Ma T F. Existence results for a model of nonlinear beam on elastic bearings. Appl Math Lett, 2000, 13:11-15
[5] Ma T F. Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. Appl Numer Math, 2003, 47:189-196
[6] Wang F l, An Y K. Existence and multiplicity of solutions for a fourth-order elliptic equation. Bound Value Probl, 2012:6
[7] Wang F l, Avci M, An Y K. Existence of solutions for fourth order elliptic equations of Kirchhoff type. J Math Anal Appl, 2014, 409:140-146
[8] Xu L P, Chen H B. Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory. Boundary Value Problems 2014, 2014:212
[9] Wu Y Z, Huang Y S, Liu Z. Sign-changing solutions for Schrödunger equations with vanishing and signchanging potentials. Acta Mathematica Scientia, 2014, 34B(3):691-702
[10] Kang D S, Luo J, Shi X L. Solutions to elliptic systems involving doubly critical nonlinearities and Hardtype potentials. Acta Mathematica Scientia, 2015, 35B(2):423-438
[11] Zou W M, Schechter M. Critical Point Theory and its Applications. New York:Springer, 2006
[12] Willem M. Minimax Theorem. Boston, MA:Birkhäuser Boston Inc, 1996
[13] Rabinowitz P H. Minimax Methods in Critical Point Theory with Application to Differential Equations. CBMS Reg Conf Ser Math, Vol 65. Providence, RI:American Mathematical Society, 1986
[14] Zou W M. Variant fountain theorems and their applications. Manuscripta Math, 2001, 104:343-358
/
| 〈 |
|
〉 |