Critical points approaches to a nonlocal elliptic problem driven by 𝑝(𝑥)-biharmonic operator

References

[1] S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), no. 3, 515–545. 10.1016/j.na.2004.09.026Suche in Google Scholar

[2] G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), no. 12, 3031–3059. 10.1016/j.jde.2008.02.025Suche in Google Scholar

[3] G. Bonanno and G. D’Aguì, Multiplicity results for a perturbed elliptic Neumann problem, Abstr. Appl. Anal. 2010 (2010), Article ID 564363. 10.1155/2010/564363Suche in Google Scholar

[4] G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1–10. 10.1080/00036810903397438Suche in Google Scholar

[5] G. D’Aguì, S. Heidarkhani and G. Molica Bisci, Multiple solutions for a perturbed mixed boundary value problem involving the one-dimensional 𝑝-Laplacian, Electron. J. Qual. Theory Differ. Equ. 2013 (2013), Paper No. 24. Suche in Google Scholar

[6] G. Dai and J. Wei, Infinitely many non-negative solutions for a p ⁢ ( x ) -Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Anal. 73 (2010), no. 10, 3420–3430. 10.1016/j.na.2010.07.029Suche in Google Scholar

[7] X. Fan and C. Ji, Existence of infinitely many solutions for a Neumann problem involving the p ⁢ ( x ) -Laplacian, J. Math. Anal. Appl. 334 (2007), no. 1, 248–260. 10.1016/j.jmaa.2006.12.055Suche in Google Scholar

[8] X. Fan and D. Zhao, On the spaces L p ⁢ ( x ) ⁢ ( Ω ) and W m , p ⁢ ( x ) ⁢ ( Ω ) , J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. 10.1006/jmaa.2000.7617Suche in Google Scholar

[9] J. R. Graef, S. Heidarkhani and L. Kong, A variational approach to a Kirchhoff-type problem involving two parameters, Results Math. 63 (2013), no. 3–4, 877–889. 10.1007/s00025-012-0238-xSuche in Google Scholar

[10] M. K. Hamdani, A. Harrabi, F. Mtiri and D. D. Repovš, Existence and multiplicity results for a new p ⁢ ( x ) -Kirchhoff problem, Nonlinear Anal. 190 (2020), Article ID 111598. 10.1016/j.na.2019.111598Suche in Google Scholar

[11] S. Heidarkhani, G. A. Afrouzi, M. Ferrara and S. Moradi, Variational approaches to impulsive elastic beam equations of Kirchhoff type, Complex Var. Elliptic Equ. 61 (2016), no. 7, 931–968. 10.1080/17476933.2015.1131681Suche in Google Scholar

[12] S. Heidarkhani, G. A. Afrouzi, S. Moradi and G. Caristi, A variational approach for solving p ⁢ ( x ) -biharmonic equations with Navier boundary conditions, Electron. J. Differential Equations 2017 (2017), Paper No. 25. Suche in Google Scholar

[13] S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi and B. Ge, Existence of one weak solution for p ⁢ ( x ) -biharmonic equations with Navier boundary conditions, Z. Angew. Math. Phys. 67 (2016), no. 3, Article ID 73. 10.1007/s00033-016-0668-5Suche in Google Scholar

[14] S. Heidarkhani, A. L. A. De Araujo, G. A. Afrouzi and S. Moradi, Multiple solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions, Math. Nachr. 291 (2018), no. 2–3, 326–342. 10.1002/mana.201600425Suche in Google Scholar

[15] S. Heidarkhani, A. L. A. De Araujo, G. A. Afrouzi and S. Moradi, Existence of three weak solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions, Fixed Point Theory 21 (2020), no. 2, 525–547. 10.24193/fpt-ro.2020.2.38Suche in Google Scholar

[16] S. Heidarkhani, M. Ferrara, A. Salari and G. Caristi, Multiplicity results for p ⁢ ( x ) -biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 61 (2016), no. 11, 1494–1516. 10.1080/17476933.2016.1182520Suche in Google Scholar

[17] S. Heidarkhani, S. Moradi and S. A. Tersian, Three solutions for second-order boundary-value problems with variable exponents, Electron. J. Qual. Theory Differ. Equ. 2018 (2018), Paper No. 33. 10.14232/ejqtde.2018.1.33Suche in Google Scholar

[18] E. M. Hssini, M. Massar and N. Tsouli, Existence and multiplicity of solutions for a p ⁢ ( x ) -Kirchhoff type problems, Bol. Soc. Parana. Mat. (3) 33 (2015), no. 2, 201–215. 10.5269/bspm.v33i2.24307Suche in Google Scholar

[19] L. Kong, Multiple solutions for fourth order elliptic problems with p ⁢ ( x ) -biharmonic operators, Opuscula Math. 36 (2016), no. 2, 253–264. 10.7494/OpMath.2016.36.2.253Suche in Google Scholar

[20] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578. 10.1137/1032120Suche in Google Scholar

[21] F.-F. Liao, S. Heidarkhani and S. Moradi, Multiple solutions for nonlocal elliptic problems driven by p ⁢ ( x ) -biharmonic operator, AIMS Math. 6 (2021), no. 4, 4156–4172. 10.3934/math.2021246Suche in Google Scholar

[22] M. Massar, E. 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. 10.22436/jmcs.08.01.04Suche in Google Scholar

[23] Q. Miao, Multiple solutions for nonlocal elliptic systems involving p ⁢ ( x ) -Biharmonic operator, Mathematics 7 (2019), no. 8, Paper No. 756. 10.3390/math7080756Suche in Google Scholar

[24] M. Mihăilescu, Existence and multiplicity of solutions for a Neumann problem involving the p ⁢ ( x ) -Laplace operator, Nonlinear Anal. 67 (2007), no. 5, 1419–1425. 10.1016/j.na.2006.07.027Suche in Google Scholar

[25] G. Molica Bisci and V. D. Rădulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math. 17 (2015), no. 1, Article ID 1450001. 10.1142/S0219199714500011Suche in Google Scholar

[26] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010), no. 4, 543–549. 10.1007/s10898-009-9438-7Suche in Google Scholar

[27] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000. 10.1007/BFb0104029Suche in Google Scholar

[28] J. Simon, Régularité de la solution d’une équation non linéaire dans R N , Journées d’Analyse Non Linéaire (Besançon 1977), Lecture Notes in Math. 665, Springer, Berlin (1978), 205–227. 10.1007/BFb0061807Suche in Google Scholar

[29] L. Vilasi, Eigenvalue estimates for stationary p ⁢ ( x ) -Kirchhoff problems, Electron. J. Differential Equations 2016 (2016), Paper No. 186. Suche in Google Scholar

[30] H. Yin and Y. Liu, Existence of three solutions for a Navier boundary value problem involving the p ⁢ ( x ) -biharmonic, Bull. Korean Math. Soc. 50 (2013), no. 6, 1817–1826. 10.4134/BKMS.2013.50.6.1817Suche in Google Scholar

[31] H. Yin and M. Xu, Existence of three solutions for a Navier boundary value problem involving the p ⁢ ( x ) -biharmonic operator, Ann. Polon. Math. 109 (2013), no. 1, 47–58. 10.4064/ap109-1-4Suche in Google Scholar

[32] A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces, Nonlinear Anal. 69 (2008), no. 10, 3629–3636. 10.1016/j.na.2007.10.001Suche in Google Scholar

[33] E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B, Springer, New York, 1990. 10.1007/978-1-4612-0981-2Suche in Google Scholar