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

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences, Farhangian University, Tehran, Iran


In this article, we study the nonlocal $p(x)$-Laplacian problem of the following form
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,
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.


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Volume 14, Issue 8
August 2023
Pages 237-247
  • Receive Date: 05 March 2022
  • Revise Date: 17 July 2022
  • Accept Date: 03 September 2022