On a new \(p(x)\)-Kirchhoff type problems with \(p(x)\)-Laplacian-like operators and Neumann boundary conditions. (English) Zbl 1535.35085
Summary: In this paper we study a Neumann boundary value problem of a new \(p(x)\)-Kirchhoff type problems driven by \(p(x)\)-Laplacian-like operators. Using the theory of variable exponent Sobolev spaces and the method of the topological degree for a class of demicontinuous operators of generalized (\(S_+\)) type, we prove the existence of a weak solutions of this problem. Our results are a natural generalisation of some existing ones in the context of \(p(x)\)-Kirchhoff type problems.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | C. Allalou, M. El Ouaarabi and S. Melliani, Existence and uniqueness results for a class of p(x)-Kirchhoff-type problems with convection term and Neumann boundary data, J. Elliptic Parabol. Equ., 8 (2022), no. 1, 617-633. · Zbl 1491.35203 |
| [2] | J. Berkovits, Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J. Differ. Equ., 234 (2007), 289-310. · Zbl 1114.47049 |
| [3] | C. Corsato, C. De Coster, F. Obersnel and P. Omari, Qualitative analysis of a curvature equation modeling MEMS with vertical loads, Nonlinear Anal. Real World Appl., 55 (2020), 103-123. · Zbl 1454.35163 |
| [4] | G. Dai and R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl., 359 (2009), 275-284. · Zbl 1172.35401 |
| [5] | M. El Ouaarabi, C. Allalou and S. Melliani, Existence result for Neumann problems with p(x)-Laplacian-like operators in generalized Sobolev spaces, Rend. Circ. Mat. Palermo, II. Ser, (2022). doi:10.1007/s12215-022-00733-y. · Zbl 1510.35130 |
| [6] | M. El Ouaarabi, C. Allalou and S. Melliani, On a class of p(x)-Laplacian-like Dirichlet problem depending on three real parameters, Arab. J. Math., 11 (2022), no. 4, 227-239. · Zbl 1497.35273 |
| [7] | M. El Ouaarabi, C. Allalou and S. Melliani, Weak solution of a Neumann boundary value problem with p(x)-Laplacian-like operator, Analysis, 42 (2022), no. 4, 271-280. · Zbl 1501.35237 |
| [8] | M. El Ouaarabi, C. Allalou and S. Melliani, Existence of weak solutions for p(x)-Laplacian-like problem with p(x)-Laplacian operator under Neumann boundary condition, S˜ao Paulo J. Math. Sci., (2022). doi:10.1007/s40863-022-00321-z. · Zbl 1501.35237 |
| [9] | M. El Ouaarabi, C. Allalou and S. Melliani, Existence of weak solution for a class of p(x)-Laplacian problems depending on three real parameters with Dirichlet condition, Bol. Soc. Mat. Mex., 28 (2022), no. 2, 31. · Zbl 1489.35144 |
| [10] | M. El Ouaarabi, C. Allalou and S. Melliani, Existence result for a Neumann boundary value problem governed by a class of p(x)-Laplacian-like equation, Asymptot. Anal., (2022). doi:10.3233/ASY-221791. · Zbl 1530.35216 |
| [11] | M. El Ouaarabi, C. Allalou and S. Melliani, On a class of nonlinear degenerate elliptic equations in weighted Sobolev spaces, Georgian Mathematical Journal, 30 (2023), no. 1, 81-94. · Zbl 1511.35157 |
| [12] | X.L. Fan and D. Zhao, On the Spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424-446. · Zbl 1028.46041 |
| [13] | R. Finn, Equilibrium Capillary Surfaces, vol. 284, Springer-Verlag, New York, 2012. · Zbl 0583.35002 |
| [14] | I.S. Kim and S.J. Hong, A topological degree for operators of generalized (S+) type, Fixed Point Theory and Appl., 1 (2015), 1-16. · Zbl 1361.47018 |
| [15] | G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. |
| [16] | O. Kováčik and J. Rákosník, On spaces L^p(x) and W^1,p(x), Czechoslovak Math. J., 41 (1991), no. 4, 592-618. · Zbl 0784.46029 |
| [17] | E. C. Lapa, V. P. Rivera and J. Q. Broncano, No-flux boundary problems involving p(x)-Laplacian-like operators, Electron.J.Diff.Equ., 2015 (2015), no. 219, 1-10. · Zbl 1330.35190 |
| [18] | W. M. Ni and J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Conveg. Lincei., 77 (1986), 231-257. |
| [19] | M. E. Ouaarabi, A. Abbassi and C. Allalou, Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces, J. Elliptic Parabol Equ., 7 (2021), no. 1, 221-242. · Zbl 1472.35184 |
| [20] | M. E. Ouaarabi, A. Abbassi and C. Allalou, Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data, Int. J. Nonlinear Anal. Appl., 13 (2021), no. 1, 2635-2653. |
| [21] | V. D. Rˇadulescu and D. D. Repoveš, Partial Differential Equations with Variable Exponents, Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, 2015. · Zbl 1343.35003 |
| [22] | M. A. Ragusa and A. Tachikawa, On continuity of minimizers for certain quadratic growth functionals, Journal of the Mathematical Society of Japan, 57 (2005), no. 3, 691-700. · Zbl 1192.49043 |
| [23] | M. A. Ragusa and A. Tachikawa, Regularity of Minimizers of some Variational Integrals with Discontinuity, Zeitschrift für Analysis und ihre Anwendungen, 27 (2008), no. 4, 469-482. · Zbl 1153.49036 |
| [24] | K. R. Rajagopal and M. Růžička, Mathematical modeling of electrorheo-logical materials, Continuum mechanics and thermodynamics, 13 (2001), no. 1, 59-78. · Zbl 0971.76100 |
| [25] | E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, 1990. · Zbl 0684.47029 |
| [26] | V. V. E. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 50 (1986), no. 4, 675-710. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
