A variational approach for solving nonlocal elliptic problems driven by a \(p(x)\)-biharmonic operator. (English) Zbl 1505.35130
Summary: This paper analyses nonlocal elliptic problems driven by a \(p(x)\)-biharmonic operator. The proof of the result is made by variational methods. One example is presented to demonstrate the application of our main results.
MSC:
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | G. A. Afrouzi, A. Hadjian, S. Heidarkhani, Steklov problem involving the p(x)-Laplacian, Electronic Journal of Differential Equations 2014 (2014), no. 134, 1-11. · Zbl 1298.35073 |
| [2] | G. A. Afrouzi, A. Hadjian, G. Molica Bisci, Some remarks for one-dimensional mean curvature problems through a local minimization principle, Advances in Nonlinear Analysis 2 (2013), no. 4, 427-441. · Zbl 1283.34014 |
| [3] | M. Bohner, G. Caristi, S. Heidarkhani, S. Moradi, A critical point approach to boundary value problems on the real line, Applied Mathematics Letters 76 (2018), 215-220. · Zbl 1388.35063 |
| [4] | G. Bonanno, G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Boundary Value Problem 2009 (2009), article number: 670675, 20 pages. · Zbl 1177.34038 |
| [5] | F. Cammaroto, A. Chinn��, B. Di Bella, Multiple solutions for a Neumann problem involving the p(x)-Laplacian, Nonlinear Analysis: Theory, Methods & Applications 71 (2009), no. 10, 4486-4492. · Zbl 1167.35381 |
| [6] | F. Cammaroto, L. Vilasi, Sequences of weak solutions for a Navier problem driven by the p(x)-biharmonic operator, Minimax Theory and its Applications 4 (2019), no. 1, 71-85. · Zbl 1415.35111 |
| [7] | N. T. Chung, Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electronic Journal of Qualitative Theory of Differential Equations 2012 (2012), no. 42, 1-13. · Zbl 1340.35039 |
| [8] | G. W. Dai, R. F. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, Journal of Mathematical Analysis and Applications 359 (2009), no. 1, 275-284. · Zbl 1172.35401 |
| [9] | G. W. Dai, J. Wei, Infinitely many non-negative solutions for a p(x)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Analysis: Theory, Methods & Applications 73 (2010), no. 10, 3420-3430. · Zbl 1201.35181 |
| [10] | S. G. Deng, A local mountain pass theorem and applications to a double perturbed p(x)-Laplacian equations, Applied Mathematics and Computation 211 (2009), no. 1, 234-241. · Zbl 1173.35045 |
| [11] | X. Fan, D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), Journal of Mathematical Analysis and Applications 263 (2001), no. 2, 424-446. · Zbl 1028.46041 |
| [12] | M. Galewski, G. Molica Bisci, Existence results for one-dimensional fractional equations, Mathematical Methods in the Applied Sciences 39 (2016), no. 6, 1480-1492. · Zbl 1381.34012 |
| [13] | B. Ge, Q. M. Zhou, Y. H. Wu, Eigenvalues of the p(x)-biharmonic operator with indefinite weight, Zeitschrift für angewandte Mathematik und Physik 66 (2015), no. 3, 1007-1021. · Zbl 1319.35147 |
| [14] | J. R. Graef, S. Heidarkhani, L. Kong, A variational approach to a Kirchhoff-type problem involving two parameters, Results in Mathematics 63 (2013), no. 63, 877-889. · Zbl 1275.35108 |
| [15] | M. K. Hamdani, A. Harrabi, F. Mtiri, D. D. Repovš, Existence and multiplicity results for a new p(x)-Kirchhoff problem, Nonlinear Analysis 190 (2020), article: 111598. · Zbl 1433.35093 |
| [16] | M. Abolghasemi and S. Moradi, J. Nonl. Evol. Equ. Appl. 2022 (2022) 89-103 · Zbl 1505.35130 |
| [17] | S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi, A variational approach for solving p(x)-biharmonic equations with Navier boundary conditions, Electronic Journal of Differential Equations 2017 (2017), no. 25, 1-15. · Zbl 1381.35036 |
| [18] | S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi, B. Ge, Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions, Zeitschrift für angewandte Mathematik und Physik 67 (2016), no. 3, 1-13. · Zbl 1353.35153 |
| [19] | S. Heidarkhani, A. L. A. de Araujo, G. A. Afrouzi, 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, 525-548. · Zbl 1455.35091 |
| [20] | S. Heidarkhani, A. L. A. de Araujo, A. Salari, Infinitely many solutions for nonlocal prob-lems with variable exponent and nonhomogeneous Neumann conditions, Boletim Sociedade Paranaense de Matematica 38 (2020), no. 4, 71-96. · Zbl 1431.35025 |
| [21] | S. Heidarkhani, A. L. A. De Araujo, G. A. Afrouzi, S. Moradi, Multiple solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions, Mathematis-che Nachrichten 91 (2018), no. 2-3, 326-342. · Zbl 1390.35083 |
| [22] | S. Heidarkhani, M. Ferrara, A. Salari, G. Caristi, Multiplicity results for p(x)-biharmonic equations with Navier boundary, Complex Variables and Elliptic Equations 61 (2016), no. 11, 1494-1516. · Zbl 1347.35104 |
| [23] | S. Heidarkhani, Y. Zhou, G. Caristi , G. A. Afrouzi, S. Moradi, Existence results for fractional differential systems through a local minimization principle, PanAmerican Mathematical Journal, to appear. |
| [24] | S. Heidarkhani, B. Ge, Critical points approaches to elliptic problems driven by a p(x)-Laplacian, Ukrainian Mathematical Journal 66 (2015), no. 12, 1883-1903. · Zbl 1359.35075 |
| [25] | S. Heidarkhani, S. Moradi, M. Avci, Critical points approaches to a nonlocal elliptic problem driven by p(x)-biharmonic operator, Georgian Mathematical Journal 29 (2022), no. 1, 55-69. · Zbl 1484.35179 |
| [26] | S. Heidarkhani, S. Moradi, D. Barilla, Existence results for second-order boundary-value problems with variable exponents, Nonlinear Analysis: Real World Applications 44 (2018), 40-53. · Zbl 1404.34029 |
| [27] | M. Hssini, M. Massar, N. Tsouli, Existence and multiplicity of solutions for a p(x)-Kirchhoff type problems, Boletim da Sociedade Paranaense de Matemática 33 (2015), no. 2, 201-215. · Zbl 1412.35004 |
| [28] | G. Kirchhoff, Vorlesungen uber mathematische Physik: Mechanik, Teubner, Leipzig, 1883. |
| [29] | L. Kong, Eigenvalues for a fourth order elliptic problem, Proceedings of the American Mathematical Society 143 (2015), no. 1, 249-258. · Zbl 1317.35166 |
| [30] | L. Kong, Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators, Opuscula Mathematica 36 (2016), no. 2, 253-264. · Zbl 1339.35130 |
| [31] | F. F. Liao, S. Heidarkhani, S. Moradi, Multiple solutions for nonlocal elliptic problems driven by p(x)-biharmonic operator, AIMS Mathematics 6 (2021), no. 4, 4156-4172. · Zbl 1525.35091 |
| [32] | J. L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, 1977). North-Holland Math. Stud, vol. 30, pp. 284-346, 1978. · Zbl 0404.35002 |
| [33] | M. Massar, N. Tsouli, M. Talbi, Multiple solutions for nonlocal system of (p(x), q(x)) Kirchhoff type, Mathematics and Computation 242 (2014), 216-226. · Zbl 1334.35051 |
| [34] | Q, Miao, Multiple Solutions for nonlinear Navier boundary systems involving (p 1 (x), . . . , p n (x)) biharmonic problem, Discrete Dynamics in Nature and Society 2016 (2016), 1-10. · Zbl 1418.35140 |
| [35] | Q, Miao, Multiple solutions for nonlocal elliptic systems involving p(x)-Biharmonic operator, Mathematics 7 (2019), no. 8, 756, 10 pages. |
| [36] | M. Ružička, Electro-rheological fluids: Modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1784, Springer, Berlin, 2000. · Zbl 0968.76531 |
| [37] | B. Ricceri, A general variational principle and some of its applications, Journal of Computa-tional and Applied Mathematics 113 (2000), no. 1-2, 401-410. · Zbl 0946.49001 |
| [38] | B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, Journal of Global Optimization 46 (2010), 543-549. · Zbl 1192.49007 |
| [39] | Z. Shen, C. Qian, Infinitely many solutions for a Kirchhoff-type problem with non-standard growth and indefinite weight, Zeitschrift für angewandte Mathematik und Physik 66 (2015), no. 2, 399-415. · Zbl 1323.35017 |
| [40] | H. Yin, Y. Liu, Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic, Bulletin of the Korean Mathematical Society 6 (2013), no. 6, 1817-1826. · Zbl 1283.35050 |
| [41] | H. Yin, M. Xu, Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic operator, Annales Polonici Mathematici 109 (2013), no. 1 , 47-54. · Zbl 1288.35247 |
| [42] | A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Analysis: Theory, Methods & Applications 69 (2008), no. 10, 3629-3636. · Zbl 1153.26312 |
| [43] | V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. · Zbl 0838.35001 |
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