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doi: 10.3934/dcdss.2022129
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## Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms

 1 Science and Technology for Defense Lab LR19DN01, CMR, Military Academy, Tunis, Tunisia, Military Aeronautical Specialities School, Sfax, Tunisia, Department of Mathematics, Faculty of Science, University of Sfax, Sfax, Tunisia 2 Department of Mathematics, Faculty of Science, University of El Manar, Tunis, Tunisia 3 Department of Mathematics, FSTH, Abdelmalek Essaadi University, Tétouan, Morocco 4 Faculty of Education, University of Ljubljana, Ljubljana, Slovenia, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia, Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

* Corresponding author: Dušan D. Repovš

Received  February 2022 Revised  May 2022 Early access June 2022

We study a class of $p(x)$-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.

Citation: Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui, Omar Darhouche, Dušan D. Repovš. Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022129
##### References:
 [1] M. Allaoui, Existence results for a class of $p(x)$-Kirchhoff problems, Studia Sci. Math. Hungar., 54 (2017), 316-331.  doi: 10.1556/012.2017.54.3.1369. [2] M. Allaoui and A. Ourraoui, Existence results for a class of $p(x)$-Kirchhoff problem with a singular weight, Mediterr. J. Math., 13 (2016), 677-686.  doi: 10.1007/s00009-015-0518-2. [3] C. O. Alves and T. Boudjeriou, Existence of solution for a class of heat equation involving the $p(x)$-Laplacian with triple regime, Z. Angew. Math. Phys., 72 (2021), Paper No. 2, 18 pp. doi: 10.1007/s00033-020-01430-5. [4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [5] A. Bahrouni, V. D. Rădulescu and P. Winkert, A critical point theorem for perturbed functionals and low perturbations of differential and nonlocal systems, Adv. Nonlinear Stud., 20 (2020), 663-674.  doi: 10.1515/ans-2020-2095. [6] C. J. Batkam, An elliptic equation under the effect of two nonlocal terms, Math. Methods Appl. Sci., 39 (2016), 1535-1547.  doi: 10.1002/mma.3587. [7] C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, arXiv: 1501.05733 [math.AP] [8] Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.  doi: 10.1088/0951-7715/28/7/2247. [9] J. Chabrowski, On bi-nonlocal problem for elliptic equations with Neumann boundary conditions, J. Anal. Math., 134 (2018), 303-334.  doi: 10.1007/s11854-018-0011-5. [10] D. Choudhuri, Existence and Hölder regularity of infinitely many solutions to a p-Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition, Z. Angew. Math. Phys., 72 (2021), Paper No. 36, 26 pp. doi: 10.1007/s00033-020-01464-9. [11] D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65-74.  doi: 10.1512/iumj.1973.22.22008. [12] F. J. S. A. Corrêa and A. C. D. R. Costa, A variational approach for a bi-nonlocal elliptic problem involving the $p(x)$-Laplacian and non-linearity with non-standard growth, Glasg. Math. J., 56 (2014), 317-333.  doi: 10.1017/S001708951300027X. [13] F. J. S. A. Corrêa and A. C. D. R. Costa, On a bi-non-local $p(x)$-Kirchhoff equation via Krasnoselskii's genus, Math. Meth. Appl. Sci., 38 (2015), 87-93.  doi: 10.1002/mma.3051. [14] F. J. S. A. Corrêa and G. M. Figueiredo, Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation, Adv. Differential Equations, 18 (2013), 587-608. [15] X.-L. Fan and Q.-H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5. [16] M. K. Hamdani, On a nonlocal asymmetric Kirchhoff problem, Asian-Eur. J. Math., 13 (2020), 2030001, 15 pp.. doi: 10.1142/S1793557120300018. [17] 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), 111598, 15 pp. doi: 10.1016/j.na.2019.111598. [18] M. K. Hamdani and D. D. Repovš, Existence of solutions for systems arising in electromagnetism, J. Math. Anal. Appl., 486 (2020), 123898, 18 pp. doi: 10.1016/j.jmaa.2020.123898. [19] Y. Jalilian, Infinitely many solutions for a bi-nonlocal equation with sign changing weight functions, Bull. Iranian Math. Soc., 42 (2016), 611-626. [20] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [21] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. [22] J.-L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. [23] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6. [24] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., 1986. doi: 10.1090/cbms/065. [25] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Boca Raton, CRC Press, 2015. doi: 10.1201/b18601. [26] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second ed., Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1. [27] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [28] Z. Yucedag, Existence of solutions for $p(x)$-Laplacian equations without Ambrosetti-Rabinowitz type condition, Bull. Malays. Math. Sci. Soc., 38 (2015), 1023-1033.  doi: 10.1007/s40840-014-0057-1. [29] B. L. Zhang, B. Ge and X.-F. Cao, Multiple Solutions for a Class of New p(x)-Kirchhoff Problem without the Ambrosetti-Rabinowitz conditions, Mathematics, 8 (2020), 2068. [30] J. Zuo, A. Fiscella and A. Bahrouni, Existence and multiplicity results for $p(\cdot)$ & $q(\cdot)$ fractional Choquard problems with variable order, Complex Var. Elliptic Equ., 67 (2022), 500-516.  doi: 10.1080/17476933.2020.1835878.

show all references

##### References:
 [1] M. Allaoui, Existence results for a class of $p(x)$-Kirchhoff problems, Studia Sci. Math. Hungar., 54 (2017), 316-331.  doi: 10.1556/012.2017.54.3.1369. [2] M. Allaoui and A. Ourraoui, Existence results for a class of $p(x)$-Kirchhoff problem with a singular weight, Mediterr. J. Math., 13 (2016), 677-686.  doi: 10.1007/s00009-015-0518-2. [3] C. O. Alves and T. Boudjeriou, Existence of solution for a class of heat equation involving the $p(x)$-Laplacian with triple regime, Z. Angew. Math. Phys., 72 (2021), Paper No. 2, 18 pp. doi: 10.1007/s00033-020-01430-5. [4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [5] A. Bahrouni, V. D. Rădulescu and P. Winkert, A critical point theorem for perturbed functionals and low perturbations of differential and nonlocal systems, Adv. Nonlinear Stud., 20 (2020), 663-674.  doi: 10.1515/ans-2020-2095. [6] C. J. Batkam, An elliptic equation under the effect of two nonlocal terms, Math. Methods Appl. Sci., 39 (2016), 1535-1547.  doi: 10.1002/mma.3587. [7] C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, arXiv: 1501.05733 [math.AP] [8] Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.  doi: 10.1088/0951-7715/28/7/2247. [9] J. Chabrowski, On bi-nonlocal problem for elliptic equations with Neumann boundary conditions, J. Anal. Math., 134 (2018), 303-334.  doi: 10.1007/s11854-018-0011-5. [10] D. Choudhuri, Existence and Hölder regularity of infinitely many solutions to a p-Kirchhoff type problem involving a singular nonlinearity without the Ambrosetti-Rabinowitz (AR) condition, Z. Angew. Math. Phys., 72 (2021), Paper No. 36, 26 pp. doi: 10.1007/s00033-020-01464-9. [11] D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65-74.  doi: 10.1512/iumj.1973.22.22008. [12] F. J. S. A. Corrêa and A. C. D. R. Costa, A variational approach for a bi-nonlocal elliptic problem involving the $p(x)$-Laplacian and non-linearity with non-standard growth, Glasg. Math. J., 56 (2014), 317-333.  doi: 10.1017/S001708951300027X. [13] F. J. S. A. Corrêa and A. C. D. R. Costa, On a bi-non-local $p(x)$-Kirchhoff equation via Krasnoselskii's genus, Math. Meth. Appl. Sci., 38 (2015), 87-93.  doi: 10.1002/mma.3051. [14] F. J. S. A. Corrêa and G. M. Figueiredo, Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation, Adv. Differential Equations, 18 (2013), 587-608. [15] X.-L. Fan and Q.-H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem, Nonlinear Anal., 52 (2003), 1843-1852.  doi: 10.1016/S0362-546X(02)00150-5. [16] M. K. Hamdani, On a nonlocal asymmetric Kirchhoff problem, Asian-Eur. J. Math., 13 (2020), 2030001, 15 pp.. doi: 10.1142/S1793557120300018. [17] 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), 111598, 15 pp. doi: 10.1016/j.na.2019.111598. [18] M. K. Hamdani and D. D. Repovš, Existence of solutions for systems arising in electromagnetism, J. Math. Anal. Appl., 486 (2020), 123898, 18 pp. doi: 10.1016/j.jmaa.2020.123898. [19] Y. Jalilian, Infinitely many solutions for a bi-nonlocal equation with sign changing weight functions, Bull. Iranian Math. Soc., 42 (2016), 611-626. [20] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [21] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. [22] J.-L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. [23] N. S. Papageorgiou, V. D. Rădulescu and D. D. Repovš, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6. [24] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., 1986. doi: 10.1090/cbms/065. [25] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Boca Raton, CRC Press, 2015. doi: 10.1201/b18601. [26] M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second ed., Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1. [27] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [28] Z. Yucedag, Existence of solutions for $p(x)$-Laplacian equations without Ambrosetti-Rabinowitz type condition, Bull. Malays. Math. Sci. Soc., 38 (2015), 1023-1033.  doi: 10.1007/s40840-014-0057-1. [29] B. L. Zhang, B. Ge and X.-F. Cao, Multiple Solutions for a Class of New p(x)-Kirchhoff Problem without the Ambrosetti-Rabinowitz conditions, Mathematics, 8 (2020), 2068. [30] J. Zuo, A. Fiscella and A. Bahrouni, Existence and multiplicity results for $p(\cdot)$ & $q(\cdot)$ fractional Choquard problems with variable order, Complex Var. Elliptic Equ., 67 (2022), 500-516.  doi: 10.1080/17476933.2020.1835878.
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