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Existence of solution for a $\varphi(\chi)$-Kirchhoff equation by Neumann condition | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 06 مرداد 1404 اصل مقاله (413.94 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2024.35153.5253 | ||
| نویسندگان | ||
| Abolfazl Sadeghi1؛ Ghasem Alizadeh Afrouzi* 1؛ Maryam Mirzapour2 | ||
| 1Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran | ||
| 2Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran | ||
| تاریخ دریافت: 13 شهریور 1403، تاریخ بازنگری: 03 مهر 1403، تاریخ پذیرش: 10 مهر 1403 | ||
| چکیده | ||
| The present article deals with a variational method, named the Mountain Pass Theorem. We prove the existence of nontrivial weak solutions for the problem of the following form \begin{align*} \begin{cases} -(\alpha-\beta \int_\Omega \dfrac{1}{\varphi(\chi)} | \nabla \upsilon|^{\varphi(\chi)} d \chi) \Delta_{\varphi(\chi)} \upsilon+ |\upsilon|^{\psi(\chi)-2}\upsilon= \lambda~ \eta(\chi,\upsilon)& \chi \in\Omega,\\ (\alpha-\beta \int_{\partial \Omega} \dfrac{1}{\varphi(\chi)} | \nabla \upsilon|^{\varphi(\chi)} d \chi) | \nabla \upsilon|^{\varphi(\chi)-2} \dfrac{\partial \upsilon}{\partial \nu}=0 & \chi \in \partial \Omega, \end{cases} \end{align*} where $\alpha \ge \beta>0, \Delta_{\varphi(\chi)} \upsilon$ is the $\varphi(\chi)$-Laplacian operator, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$ and $\nu$ is the outer unit normal to $\partial \Omega$, $\varphi(\chi), \psi(\chi) \in C(\bar{\Omega})$ with $1< \varphi(\chi)<N, \varphi(\chi)<\psi(\chi)< \varphi^*(\chi):= \dfrac{N \varphi(\chi)}{N- \varphi(\chi)}$, $ \lambda>0$ is a real parameter and $\eta(\chi,t) \in C( \bar{\Omega} \times \mathbb{R}, \mathbb{R})$. | ||
| کلیدواژهها | ||
| generalized Lebesgue-Sobolev spaces؛ weak solution؛ Mountain pass theorem | ||
| مراجع | ||
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