Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 16
157
Views
5
CrossRef citations to date
0
Altmetric
Research Article

On a p⋅-biharmonic problem of Kirchhoff type involving critical growth

ORCID Icon & ORCID Icon
Pages 5700-5726 | Received 18 Dec 2020, Accepted 03 Mar 2021, Published online: 22 Mar 2021

References

  • Woinowsky-Krieger S. The effect of axial force on the vibration of hinged bars. J Appl Mech. 1950;17:35–36.
  • Kirchhoff G. Mechanik. Leipzig: Teubner; 1883.
  • Berger M. A new approach to the large deflection of plate. J Appl Mech. 1955;22:465–472.
  • Ma TF. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Discrete Contin Dyn Syst Suppl. 2007;2:145–201.
  • Wang F, An Y. Existence and multiplicity of solutions for a fourth-order elliptic equation. Bound Value Probl. 2012;2012:6.
  • Wang F, Avci M, An Y. Existence of solutions for fourth-order elliptic equations of Kirchhoff type. J Math Anal Appl. 2014;409:140–146.
  • Bae JH, Kim JM, Lee J, et al. Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations. Bound Value Probl. 2019;2019:125.
  • Ferrara M, Khademloo S, Heidarkhani S. Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl Math Comput. 2014;234:316–325.
  • Liang S, Zhang Z. Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in RN. J Math Phys. 2016;57:111505.
  • Wu DL, Li F. Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in RN. Comput Math Appl. 2020;79(2):489–499.
  • Xu L, Chen H. Multiple solutions for the nonhomogeneous fourth order elliptic equations of Kirchhoff-type. Taiwanese J Math. 2015;19(4):1215–1226.
  • Sun J, Wu TF. Multiplicity and concentration of nontrivial solutions for the generalized extensible beam equations. Electron J Differ Equ. 2019;2019(41):1–23.
  • Zhang W, Tang X, Cheng B, et al. Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type. Commun Pure Appl Anal. 2016;15(6):2161–2177.
  • Cabada A, Figueiredo GM. A generalization of an extensible beam equation with critical growth in RN. Nonlinear Anal Real World Appl. 2014;20:134–142.
  • Figueiredo GM, Nascimento RG. Multiplicity of solutions for equations involving a nonlocal term and the biharmonic operator. Electron J Differ Equ. 2016;2016(217):1–15.
  • Acerbi E, Mingione G. Regularity results for stationary electro-rheological fluids. Arch Ration Mech Anal. 2002;164:213–259.
  • Acerbi E, Mingione G, Seregin G. Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann Inst H Poincaré Anal Non Linéaire. 2004;21(1):25–60.
  • Chen Y, Levine S, Rao M. Variable exponent linear growth functionals in image restoration. SIAM J Appl Math. 2006;66(4):1383–1406.
  • Růžička M. Electrorheological fluids: modeling and mathematical theory. Berlin: Springer-Verlag; 2000. (Lecture notes in mathematics; 1748).
  • Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev spaces with variable exponents. Heidelberg: Springer-Verlag; 2011. (Lecture notes in mathematics; 2017).
  • Rădulescu V, Repov s˘ D. Partial differential equations with variable exponents: variational methods and qualitative analysis. Boca Raton (FL): CRC Press, Taylor and Francis Group; 2015.
  • Alves CO, Barreiro JLP. Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth. J Math Anal Appl. 2013;403:143–154.
  • Bonder JF, Silva A. Concentration-compactness principle for variable exponent spaces and applications. Electron J Differ Equ. 2010;2010(141):1–18.
  • Correa FJSA, Costa ACR. On a p(x)-Kirchhoff equation with critical exponent and an additional nonlocal term via truncation argument. Math Nachr. 2015;288(11–12):1226–1240.
  • Fu Y. The principle of concentration compactness in Lp(x) spaces and its application. Nonlinear Anal. 2009;71:1876–1892.
  • Ho K, Sim I. On degenerate p(x)-Laplace equations involving critical growth with two parameters. Nonlinear Anal. 2016;132:95–114.
  • Hurtado EJ, Miyagaki OH, da Silva Rodrigues R. Existence and asymptotic behaviour for a Kirchhoff type equation with variable critical growth exponent. Milan J Math. 2017;85:71–102.
  • Ayoujil A, Amrouss ARE. On the spectrum of a fourth order elliptic equation with variable exponent. Nonlinear Anal. 71(14):4916–4926.
  • Baraket S, Rădulescu V. Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent. Adv Nonlinear Stud. 2016;16(3):409–419.
  • Boureanu MM, Rădulescu V, Repov s˘ D. On a p(⋅)-biharmonic problem with no-flux boundary condition. Comput Math Appl. 2016;72(9):2505–2515.
  • Chung NT. Existence of solutions for perturbed fourth order elliptic equations with variable exponents. Electron J Qual Theory Differ Equ. 2018;2018(96):1–19.
  • Kong L. Eigenvalues for a fourth order elliptic problem. Proc Am Math Soc. 2015;143:249–258.
  • Lions P-L. The concentration-compactness principle in the calculus of variations. The limit case Part 1. Rev Mat Iberoam. 1985;1(1):145–201.
  • 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.
  • Zang A, Fu Y. Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces. Nonlinear Anal. 2008;69:3629–3636.
  • Brezis H. Functional analysis, Sobolev spaces and partial differential equations. New York (NY): Springer; 2011. (Universitext).
  • Fonseca I, Leoni G. Modern methods in the calculus of variations: Lp spaces. New York: Springer; 2007.
  • Bernis F, Azorero JG, Alonso IP. Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth-order. Adv Differ Equ. 1996;1:219–240.
  • Fan X, Zhang Q, Zhao D. Eigenvalues of p(x)-Laplacian Dirichlet problem. J Math Anal Appl. 2005;302:306–317.
  • Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. Providence (RI): American Mathematical Society; 1986. (CBMS regional conference series in mathematics; vol. 65).
  • Azorero JG, Alonso IP. Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans Am Math Soc. 1991;323(2):877–895.

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.