Infinitely many solutions to a system of double phase equations with variable exponents and Dirichlet conditions

This work is devoted to investigating a Dirichlet problem for a system of double phase equations with variable exponents. The system is defined as follows: $$ \textstyle\begin{cases} -\operatorname{div}\left (|\nabla \phi |^{\alpha _{1}(y)-2}\nabla \phi + \mu _{1}(y)|\nabla \phi |^{\beta _{1}(y)-2}\nabla \phi \right ) = f_{1}(\phi ,\varphi ) & \text{in } \Upsilon , \\ -\operatorname{div}\left (|\nabla \varphi |^{\alpha _{2}(y)-2}\nabla \varphi + \mu _{2}(y)|\nabla \varphi |^{\beta _{2}(y)-2}\nabla \varphi \right ) = f_{2}(\phi ,\varphi ) & \text{in } \Upsilon , \\ \phi = 0, \quad \varphi = 0 & \text{on } \partial \Upsilon , \end{cases} $$ where $\Upsilon \subset \mathbb{R}^{N}$ denotes a bounded domain with a smooth boundary. Using a variational principle introduced by B. Ricceri and critical point theory in the setting of variable exponent Sobolev spaces, we prove that the system admits infinitely many weak solutions.