Existence and multiplicity of solutions for Neumann boundary value problems involving nonlocal $p(x)$-Laplacian equations

	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
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
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