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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 | ||
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