پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شماره‌ها | 642 |
| تعداد مقالات | 9,403 |
| تعداد مشاهده مقاله | 68,117,175 |
| تعداد دریافت فایل اصل مقاله | 37,287,528 |
Existence and multiplicity of solutions for Neumann boundary value problems involving nonlocal $p(x)$-Laplacian equations | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 22، دوره 14، شماره 8، آبان 2023، صفحه 237-247 اصل مقاله (428.19 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.7212 | ||
| نویسنده | ||
| Maryam Mirzapour* | ||
| Department of Mathematics, Faculty of Mathematical Sciences, Farhangian University, Tehran, Iran | ||
| تاریخ دریافت: 14 اسفند 1400، تاریخ بازنگری: 26 تیر 1401، تاریخ پذیرش: 12 شهریور 1401 | ||
| چکیده | ||
| In this article, we study the nonlocal $p(x)$-Laplacian problem of the following form $$ \left\{\begin{array}{ll} M\Big (\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\Big)\Big(-\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u+|u|^{p(x)-2}u\Big) =\lambda f(x,u) & \text{ in } \Omega,\\ M\Big (\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\Big)|\nabla u|^{p(x)-2}\nabla \frac{\partial u}{\partial \nu}=\mu g(x,u) & \textrm{ on } \partial\Omega, \end{array}\right. $$ By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem. | ||
| کلیدواژه‌ها | ||
| Generalized Lebesgue-Sobolev spaces؛ Nonlocal condition؛ Mountain pass theorem؛ Fountain theorem؛ Dual fountain theorem | ||
| مراجع | ||
|
[1] G.A. Afrouzi, M. Mirzapour and N.T. Chung, Existence and multiplicity of solutions for Kirchhoff type problems involving p(・)-biharmonic operators, Z. Anal. Anwend., 33 (2014), 289–303. [2] G.A. Afrouzi, M. Mirzapour and N.T. Chung, Existence and multiplicity of solutions for a p(x)-Kirchhoff type equation, Rend. Sem. Mat. Univ. Padova 136 (2016), 95–109. [3] G.A. Afrouzi and M. Mirzapour, Eigenvalue problems for p(x)-Kirchhoff type equations, Electron. J. Differ. Equ. 253 (2013), 1–10. [4] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349–381. [5] E. Acerbi and G. Mingione, Gradient estimate for the p(x)-Laplacian system, J. Reine Angew. Math. 584 (2005), 117–148. [6] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. [7] M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), 4619-4627. [8] J.M.B. do O, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, J. Diff. Equ. 11 (1997), 1–15. [9] D.E. Edmunds, J. Rakosnık, Density of smooth functions in Wk,p(x)(Ω), Proc. R. Soc. A, 437 (1992), 229-236. [10] D.E. Edmunds and J. Rakosnık, Sobolev embedding with variable exponent, Studia Math. 143 (2000), 267–293. [11] X.L. Fan and X.Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN, Nonlinear Anal. 59 (2004), 173–188. [12] X.L. Fan and D. Zhao, On the generalized Orlicz-Sobolev spaces Wk,p(x)(Ω), J. Gansu Educ. College 12 (1998), no. 1, 1–6. [13] X.L. Fan, J.S. Shen and D. Zhao, Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), 749–760. [14] X.L. Fan and D. Zhao, On the spaces Lp(x) and Wm,p(x), J. Math. Anal. Appl., 263 (2001), 424–446. [15] X.L. Fan and Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843–1852. [16] X.L. Fan, Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl. 312 (2005), 464–477. [17] D.D. Hai and R. Shivaji, An existence result on positive solutions of p-Laplacian systems, Nonlinear Anal. 56 (2004), 1007–1010. [18] D.W. Huang and Y.Q. Li, Multiplicity of solutions for a noncooperative p-Laplacian elliptic system in RN, J. Differ. Equ. 215 (2005), no. 1, 206–223. [19] J.L. Lions, On some questions in boundary value problems of mathematical physics, Proc. Int. Symp. Continuum Mech. Partial Differ. Equ., Rio de Janeiro 1977, in: de la Penha, Medeiros (Eds.), Math. Stud., North-Holland, 30 (1978), pp. 284–346. [20] T.F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal. 63 (2005), 1967–1977. [21] S. Ma, Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups, Nonlinear Anal. 73 (2010), 3856–3872. [22] K. Perera and Z.T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221 (2006), 246–255. [23] M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. [24] T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal. 68 (2008), 1733–1745. [25] J. Yao, Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear Anal. 68 (2008), 1271–1283. [26] G. Zhang and Y. Wang, Some existence results for a class of degenerate semilinear elliptic systems, J. Math. Anal. Appl. 333 (2007), 904–918. [27] J.F. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991. [28] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9 (1987), 33–66. [29] N.B. Zographopoulos, On a class of degenerate potential elliptic system, Nonlinear Diff. Equ. Appl. 11 (2004), 191–199. [30] N.B. Zographopoulos, p-Laplacian systems on resonance, Appl. Anal. 83 (2004), 509–519. [31] N.B. Zographopoulos, On the principal eigenvalue of degenerate quasilinear elliptic systems, Math. Nachr. 281 (2008), no. 9, 1351–1365. | ||
|
آمار تعداد مشاهده مقاله: 16,549 تعداد دریافت فایل اصل مقاله: 5,964 |
||