پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شمارهها | 687 |
| تعداد مقالات | 9,985 |
| تعداد مشاهده مقاله | 70,957,078 |
| تعداد دریافت فایل اصل مقاله | 62,422,384 |
Existence of three weak solutions for an anisotropic quasi-linear elliptic problem | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 9، دوره 14، شماره 10، دی 2023، صفحه 85-93 اصل مقاله (399.56 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2023.30429.4399 | ||
| نویسندگان | ||
| Ahmed Ahmed* 1؛ Mohamed Saad Bouh Elemine Vall2 | ||
| 1Mathematics and Computer Sciences Department, University of Nouakchott, Faculty of Science and Technology, Nouakchott, Mauritania | ||
| 2Department of Mathematics, Professional University Institute, Research unity: Modelling and Scientific Calculus, Nouakchott, Mauritania | ||
| تاریخ دریافت: 02 اردیبهشت 1402، تاریخ بازنگری: 10 اردیبهشت 1402، تاریخ پذیرش: 20 اردیبهشت 1402 | ||
| چکیده | ||
| We consider in this paper a Neumann $\vec{p}(x)-$elliptic problems of the type $$\left\{\begin{array}{ll} - \Delta_{\vec{p}(x)} u+ \lambda(x)|u|^{p_{0}(x)-2}u = \alpha f(x,u)+ \beta g(x,u) \quad &\mbox{in} \quad \Omega, \\ \displaystyle\sum_{i=1}^{N}\Big| \frac{\partial u}{\partial x_{i}}\Big|^{p_{i}(x)-2}\frac{\partial u}{\partial x_{i}}\gamma_{i} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right.$$ We prove the existence of three weak solutions in the framework of anisotropic Sobolev spaces with variable exponent $W^{1,\vec{p}(\cdot)}(\Omega)$ under some hypotheses. The approach is based on a recent three critical points theorem for differentiable functionals. | ||
| کلیدواژهها | ||
| Neumann elliptic problem؛ weak solutions؛ Variational principle؛ Anisotropic variable exponent Sobolev spaces | ||
| مراجع | ||
|
[1] A. Ahmed, H. Hjiaj and A. Touzani, Existence of infinitely many weak solutions for a Neumann elliptic equations involving the ⃗p(・)-Laplacian operator, Rend. Circ. Mat. Palermo (2) 64 (2015), no. 3, 459–473. [2] G. Anello and G. Cordaro, An existence theorem for the Neumann problem involving the p-Laplacian, J. Convex. Anal. 10 (2003), no. 1, 185-198. [3] S. Antontsev and S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard conditions, Handbook of Differential Equations. Stationary Partial Differential Equations, Vol. 3, Ch. 1, Elsevier, 2006, pp. 1-100. [4] S. N. Antontsev and J. F. Rodrigues, On stationary thermorheological viscous flows, Annali dell’Universit`a di Ferrara. Sezione 7: Sci. Mate. 52 (2006), no. 1, 19-36. [5] M. Bendahmane, M. Chrif and S. El Manouni, An approximation result in generalized anisotropic Sobolev spaces and application, Z. Anal. Anwend. 30 (2011), no. 3, 341–353. [6] G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), no. 4, 651-665. [7] G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel) 80 (2003), 424–429. [8] M. M. Boureanu, A new class of nonhomogeneous differential operators and applications to anisotropic systems, Complex Var. Elliptic Equ. 61 (2016), no. 5, 712–730. [9] M. M. Boureanu, A. Matei and M. Sofonea, Nonlinear problems with p(・)-growth conditions and applications to antiplane contact models. Adv. Nonlinear Stud. 14 (2014), 295–313. [10] M. M. Boureanu and V. D. R˘adulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. 75 (2012), 4471–4482. [11] Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406. [12] G. Dai. Three solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian, Nonlinear Anal. 70 (2009), no. 10, 3755–3760. [13] L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., Springer, 2011. [14] M.S.B. Elemine Vall and A. Ahmed, Infinitely many weak solutions for perturbed nonlinear elliptic Neumann problem in Musielak-Orlicz-Sobolev framework, Acta Sci. Math. (Szeged) 86 (2020), no. 3-4, 601–616. [15] X. Fan, On nonlocal ⃗p(・)-Laplacian equations, Nonlinear Anal. 73 (2010), no. 10, 3364–3375. [16] P. Harjulehto and P. Hasto, Sobolev inequalities for variable exponents attaining the values 1 and n, Publ. Mat. 52 (2008), no. 2, 347–363. [17] S. Heidarkhani, G. Caristi and M. Ferrara, Perturbed Kirchhoff-type Neumann problems in Orlicz-Sobolev spaces, Comput. Math. Appl. 71 (2016), 2008–2019. [18] C. Ji, An eigenvalue of an anisotropic quasilinear elliptic equation with variable exponent and Neumann boundary condition, Nonlinear Anal. 71 (2009), no. 10, 4507-4514. [19] D. Liu and P. Zhao, Solutions for a quasilinear elliptic equation in Musielak-Sobolev spaces, Nonlinear Anal. Real World Appl. 26 (2015), 315-329. [20] R. Mashiyev, Three Solutions to a Neumann problem for elliptic equations with variable exponent, Arab J. Sci. Eng. 36 (2011), no. 8, 1559-1567. [21] Q. Liu, Existence of three solutions for p(x)-Laplacian equations, Nonlinear Anal. 68 (2008), no. 7, 2119–2127. [22] M. Mihailescu and G. Morosanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Appl. Anal. 89 (2010), no. 2, 257–271. [23] W.W. Pan, G.A. Afrouzi and L. Li, Three solutions to a p(x)-Laplacian problem in weighted-variable-exponent Sobolev space An. St. Univ. Ovidius Constant，a 21 (2013), no. 2, 195–205. [24] J. Rakosnık, Some remarks to anisotropic Sobolev spaces I, Beitrage zur Anal. 13 (1979), 55–68. [25] J. Rakosnık, Some remarks to anisotropic Sobolev spaces II, Beitrage zur Anal. 15 (1981), 127–140. [26] A. Razani and G. M. Figueiredo, Degenerated and competing anisotropic (p, q)-Laplacians with weights, Appl. Anal. (2022). https://doi.org/10.1080/00036811.2022.2119137. [27] A. Razani and G. M. Figueiredo, A positive solution for an anisotropic (p, q)-Laplacian, Discrete Contin. Dyn. Syst. Ser. S 16 (2023), no. 6, 1629-–1643. https://doi.org/10.3934/dcdss.2022147. [28] A. Razani and G. M. Figueiredo, Existence of infinitely many solutions for an anisotropic equation using genus theory, Math. Meth. Appl. Sci. 45 (2022), no. 12, 7591–7606. [29] A. Razani, Nonstandard competing anisotropic (p, q)-Laplacians with convolution, Bound. Value Probl. 2022 (2022), 87. [30] B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc. 33 (2001), 331–340. [31] B. Ricceri. A three critical points theorem revisited, Nonlinear Anal. 70 (2009), no. 9, 3084–3089. [32] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, 2000. [33] T. Soltani and A. Razani, Solutions for an anisotropic elliptic problem involving nonlinear terms, Quaest. Math. (2023), 1-20. https://doi.org/10.2989/16073606.2023.2190045 | ||
|
آمار تعداد مشاهده مقاله: 16,760 تعداد دریافت فایل اصل مقاله: 10,391 |
||