Research Article
BibTex RIS Cite

On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent

Year 2019, Volume: 3 Issue: 1, 35 - 45, 31.03.2019
https://doi.org/10.31197/atnaa.495567

Abstract

In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent
$$
\left\{\begin{array}{ll}
\Delta^2_{p(x)}u-M\left(\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx\right)\Delta_{p(x)} u  = \lambda f(x,u) \quad \text{ in }\Omega,\\
u=\Delta u = 0 \quad \text{ on } \partial\Omega, 
\end{array}\right.
$$
where $\Omega \subset \R^N$, $N \geq 3$, is a smooth bounded domain, $M(t)=a+bt^\kappa$, $a, \kappa>0$, $b \geq 0$, $\lambda$ is a positive parameter, $\Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u)$ is the operator of fourth order called the $p(x)$-biharmonic operator, $\Delta_{p(x)}u = \operatorname{div} \left(|\nabla u|^{p(x)-2}\nabla u\right)$ is the $p(x)$-Laplacian, $p:\overline\Omega \to \R$ is a log-H\"{o}lder continuous function and $f: \overline\Omega\times \R\to \R$ is a continuous function satisfying some certain conditions. Using Ekeland's variational principle combined with variational techniques, an existence result is established in an appropriate function space.

References

  • [1] G.A. Afrouzi, M. Mirzapour, N.T. Chung, Existence and multiplicity of solutions for Kirchhoff type problems involving p(x)-biharmonic operators, Z. Anal. Anwend., 33 (2014), 289-303.
  • [2] K.B. Ali, A. Ghanmi, K. Kefi, Minimax method involving singular p(x)-Kirchhoff equation, J. Math. Physics, 58 (2017): 111505.
  • [3] H. Ansari, S.M. Vaezpour, Existence and multiplicity of solutions for fourth-order elliptic Kirchhoff equations with potential term, Complex Var. Elliptic Equ., 60 (2015), 668-695.
  • [4] A. Ayoujil, A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent,Nonlinear Anal., 71 (2009), 4916-4926.
  • [5] J.M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.
  • [6] G. Bonanno, A. Chinni, Existence and multiplicity of weak solutions for elliptic Dirichlet problems withvariable exponent, J. Math. Anal. Appl., 418 (2014), 812-827.
  • [7] M.M. Boureanu, V. Radulescu, D. Repovs, On a (.)-biharmonic problem with no-flux boundary condition,Comput. & Math. Appl., 72 (2016), 2505-2515.
  • [8] A. Cabada, G.M. Figueiredo, A generalization of an extensible beam equation with critical growth in RN,Nonlinear Anal. Real World Appl., 20 (2014), 134-142.
  • [9] N.T Chung, Q.A. Ngo, Multiple solutions for a class of quasilinear elliptic equations of p(x)-Laplacian typewith nonlinear boundary conditions, Proc. Royal Soc. Edinburgh Sect. A: Mathematics, 140(2) (2010),259-272.
  • [10] N.T. Chung, Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Var. Elliptic Equ., 58(12) (2013), 1637-1646.
  • [11] N.T Chung, Existence of solutions for perturbed fourth order elliptic equations with variable exponents,Electron. J. Qual. Theory Differ. Equ., 2018(96) (2018), 1-19.
  • [12] F. Colasuonno, P. Pucci, Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equations, NonlinearAnal., 74 (2011), 5962-5974.
  • [13] G. Dai, Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian, Appl.Anal., 92 (2013), 191-210.
  • [14] L. Ding, L. Li, Two nontrivial solutions for the nonhomogenous fourth order Kirchhoff equation, Z. Anal.Anwend., 36 (2017), 191-207.
  • [15] L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents,Lecture Notes, vol. 2017, Springer-Verlag, Berlin, 2011.
  • [16] D.E. Edmunds, J. Rakosn´ik, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.
  • [17] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
  • [18] M. Ferrara, S. Khademloo, S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff typeelliptic problems, Appl. Math. Comput., 234 (2014), 316-325.
  • [19] A. Ghanmi, Nontrivial solutions for Kirchhoff-type problems involving the p(x)-Laplace operator, RockyMountain J. Math., 48(4) (2018), 1145-1158.
  • [20] K. Kefi, V.D. Radulescu, On a p(x)-biharmonic problem with singular weights, Z. Angew. Math. Phys., 68(2017): 80.
  • [21] L. Kong, Eigenvalues for a fourth order elliptic problem, Proc. Amer. Math. Soc., 143 (2015), 249-258.
  • [22] M. Massar, El M. Hssini, N. Tsouli, M. Talbi, Infinitely many solutions for a fourth-order Kirchhoff typeelliptic problem, J. Math. Comput. Sci., 8 (2014), 33-51.
  • [23] M. Mihailescu, V.D. Radulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaceswith variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.
  • [24] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin,1983.
  • [25] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
  • [26] Y. Song, S. Shi, Multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with criticalexponent, J. Dyn. Control Syst., 23 (2017), 375-386.
  • [27] F. Wang, M. Avci, Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J.Math. Anal. Appl., 409 (2014), 140-146.
  • [28] F. Wang, Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. ValueProbl., 2012 (2012): 6.
  • [29] A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces,Nonlinear Anal., 69 (2008), 3629-3636.
  • [30] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv.,9 (1987), 33-66.
Year 2019, Volume: 3 Issue: 1, 35 - 45, 31.03.2019
https://doi.org/10.31197/atnaa.495567

Abstract

References

  • [1] G.A. Afrouzi, M. Mirzapour, N.T. Chung, Existence and multiplicity of solutions for Kirchhoff type problems involving p(x)-biharmonic operators, Z. Anal. Anwend., 33 (2014), 289-303.
  • [2] K.B. Ali, A. Ghanmi, K. Kefi, Minimax method involving singular p(x)-Kirchhoff equation, J. Math. Physics, 58 (2017): 111505.
  • [3] H. Ansari, S.M. Vaezpour, Existence and multiplicity of solutions for fourth-order elliptic Kirchhoff equations with potential term, Complex Var. Elliptic Equ., 60 (2015), 668-695.
  • [4] A. Ayoujil, A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent,Nonlinear Anal., 71 (2009), 4916-4926.
  • [5] J.M. Ball, Initial-boundary value for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.
  • [6] G. Bonanno, A. Chinni, Existence and multiplicity of weak solutions for elliptic Dirichlet problems withvariable exponent, J. Math. Anal. Appl., 418 (2014), 812-827.
  • [7] M.M. Boureanu, V. Radulescu, D. Repovs, On a (.)-biharmonic problem with no-flux boundary condition,Comput. & Math. Appl., 72 (2016), 2505-2515.
  • [8] A. Cabada, G.M. Figueiredo, A generalization of an extensible beam equation with critical growth in RN,Nonlinear Anal. Real World Appl., 20 (2014), 134-142.
  • [9] N.T Chung, Q.A. Ngo, Multiple solutions for a class of quasilinear elliptic equations of p(x)-Laplacian typewith nonlinear boundary conditions, Proc. Royal Soc. Edinburgh Sect. A: Mathematics, 140(2) (2010),259-272.
  • [10] N.T. Chung, Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Var. Elliptic Equ., 58(12) (2013), 1637-1646.
  • [11] N.T Chung, Existence of solutions for perturbed fourth order elliptic equations with variable exponents,Electron. J. Qual. Theory Differ. Equ., 2018(96) (2018), 1-19.
  • [12] F. Colasuonno, P. Pucci, Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equations, NonlinearAnal., 74 (2011), 5962-5974.
  • [13] G. Dai, Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian, Appl.Anal., 92 (2013), 191-210.
  • [14] L. Ding, L. Li, Two nontrivial solutions for the nonhomogenous fourth order Kirchhoff equation, Z. Anal.Anwend., 36 (2017), 191-207.
  • [15] L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents,Lecture Notes, vol. 2017, Springer-Verlag, Berlin, 2011.
  • [16] D.E. Edmunds, J. Rakosn´ik, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.
  • [17] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
  • [18] M. Ferrara, S. Khademloo, S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff typeelliptic problems, Appl. Math. Comput., 234 (2014), 316-325.
  • [19] A. Ghanmi, Nontrivial solutions for Kirchhoff-type problems involving the p(x)-Laplace operator, RockyMountain J. Math., 48(4) (2018), 1145-1158.
  • [20] K. Kefi, V.D. Radulescu, On a p(x)-biharmonic problem with singular weights, Z. Angew. Math. Phys., 68(2017): 80.
  • [21] L. Kong, Eigenvalues for a fourth order elliptic problem, Proc. Amer. Math. Soc., 143 (2015), 249-258.
  • [22] M. Massar, El M. Hssini, N. Tsouli, M. Talbi, Infinitely many solutions for a fourth-order Kirchhoff typeelliptic problem, J. Math. Comput. Sci., 8 (2014), 33-51.
  • [23] M. Mihailescu, V.D. Radulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaceswith variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.
  • [24] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin,1983.
  • [25] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
  • [26] Y. Song, S. Shi, Multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with criticalexponent, J. Dyn. Control Syst., 23 (2017), 375-386.
  • [27] F. Wang, M. Avci, Y. An, Existence of solutions for fourth order elliptic equations of Kirchhoff type, J.Math. Anal. Appl., 409 (2014), 140-146.
  • [28] F. Wang, Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. ValueProbl., 2012 (2012): 6.
  • [29] A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces,Nonlinear Anal., 69 (2008), 3629-3636.
  • [30] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv.,9 (1987), 33-66.
There are 30 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Nguyen Thanh Chung 0000-0001-7345-620X

Publication Date March 31, 2019
Published in Issue Year 2019 Volume: 3 Issue: 1

Cite