On a fourth-order elliptic Kirchhoff type problem with critical Sobolev exponent

Year 2020, Volume: 4 Issue: 4, 394 - 401, 30.12.2020

Mohammed Massar

https://doi.org/10.31197/atnaa.683089

Abstract

This work is concerned with a class of fourth-order elliptic Kirchhoff type problems involving the critical term. By means of the truncation and the concentration compact argument, for each positive integer k the existence of $k$ pairs nontrivial solutions is established.

Keywords

Elliptic equations, Kirchhoff type problem, variational methods

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References

• [1] C. O. Alves, F. J. S. A. Corr\^{e}a and G.M. Figueiredo, On class of nonlocal elliptic problem with critical growth, Differ. Equ. Appl. 2(3) (2010), 409-419.
• [2] A. Azzollini, The elliptic Kirchhoff equation in R^N perturbed by a local nonlinearity, Differ. Integral Equ. 25 (2012), 543-554.
• [3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math. 36 (1983), 437-477.
• [4] M. Ferrar and S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems, Appl. Math. Comput. 234 (2014), 316-325.
• [5] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument}, J. Math. Anal. Appl. 401 (2013), 706-713.
• [6] G. M. Figueiredo and J.R.S. Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical and critical growth, Differ. Integral Equ. 25 (2012), 853-868.
• [7] M. F. Furtado, L. D. de Oliveira and J. P. P. da Silva, Multiple solutions for a Kirchhoff equation with critical growth, Z. Angew. Math. Phys. 70 (11) (2019), 1-15.
• [8] A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for p-Kirchhoff type problems with critical exponent, Electron. J. Differ. Equ. 2011 (105) (2011), 1-8.
• [9] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
• [10] H. Y. Li and J. F. Liao, Existence and multiplicity of solutions for superlinear Kirchhoff-type equations with critical Sobolev exponent in R^N, Comput. Math. Appl. 72 (2016), 2900-2907.
• [11] P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, part I, II. Rev. Mat. Iberoamericana 1 (1985), 145-201, 45-121.
• [12] M. Massar, EL. M. Hssini, N. Tsouli and M. Talbi, Infinitely many solutions for a fourth-order Kirchhoff type elliptic problem, J. Math. Comput. Sci. 8 (2014), 33-51.
• [13] A. M. Mao and J. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 (2009), 1275-1287.
• [14] D. Naimen, Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, NoDEA Nonlinear Differ. Equ. Appl. 21 (6) (2014), 885-914.
• [15] E. A. B. Silva and M. S. Xavier, Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 341-358.
• [16] J. Sun and C. Tang, Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal. 74 (2011), 1212-1222.
• [17] F. Wang and Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. Value Prob. 2012, 6 (2012).