In this paper, we consider the quasilinear elliptic equation of the form: \( \left\{ \begin{array}{ll} -\textrm{div }\left( a(x, u_{1},\nabla u_{1})\right) + |u_1|^{p(x)-2} u_1 = f_{1}(x) \quad & \text{ in } \Omega _1, \\ -\textrm{div }\left( a(x, u_{2},\nabla u_{2})\right) + |u_2|^{p(x)-2} u_2 = f_{2}(x) \quad & \text{ in } \Omega _2, \\ u_{1} = 0 & \text{ on } \partial \Omega ,\\ a(x, u_{1},\nabla u_{1}) \nu _1 = a(x,u_{2},\nabla u_{2}) \nu _1 \quad & \text{ on } \Gamma ,\\ a(x, u_{1},\nabla u_{1}) \nu _1 = -g(x) |u_1-u_2|^{p(x)-2}(u_1 - u_2) \quad & \text{ on } \Gamma , \end{array} \right. \) where \(\Omega \) is an open bounded and connected subset of \( I\!\!R^N\) ( \(N \ge 2\) ), with a Lipschitz boundary \(\partial \Omega \) , such that \(\Omega \) is decomposed as \(\Omega _1 \cup \Omega _2 \cup \Gamma \) , where \(\Omega _2\) is an open subset of \(\Omega \) such that \(\overline{\Omega _{2}} \subset \Omega \) , and \(\Omega _1 = \Omega \setminus \overline{\Omega _2}\) and \(\Gamma = \partial \Omega _{2},\) where the right-hand side \(f = (f_{1},f_{2})\) is assumed to be in the dual. We study the existence of weak solutions for the quasilinear \(p(x)-\) elliptic problem with two-component domains.