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Variational method has played an important role in solving problems of uniqueness and existence of the nonlinear works as well as analysis. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations economy, optimization, game theory and medicine. Recently, the existence of infinitely many weak solutions for some non-local problems of Kirchhoff type with Dirichlet boundary condition are studied \cite{Makvand2019}. Here, a suitable method is presented to treat the elliptic partial derivative equations, especially $(p(x),q(x))$-Laplacian-like systems. This kind of equations are used in the study of fluid flow, diffusive transport akin to diffusion, rheology, probability, electrical networks, etc. Here, the existence of infinitely many weak solutions for some boundary value problems involving the $(p(x),q(x))$-Laplacian-like operators is proved. The method is based on variational methods and critical point theory.
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