Existence of infinitely many solutions for logarithmic double-phase systems with Dirichlet conditions

Abstract This paper investigates a Dirichlet problem for a system of logarithmic double-phase equations with variable exponents. The system is given by − d i v | ∇ u | α 1 ( τ ) − 2 ∇ u + μ ( τ ) log ( e + | ∇ u | ) + | ∇ u | β 1 ( τ ) ( e + | ∇ u | ) | ∇ u | β 1 ( τ ) − 2 ∇ u = h 1 ( u , v )   in  D , − d i v | ∇ v | α 2 ( τ ) − 2 ∇ v + μ ( τ ) log ( e + | ∇ v | ) + | ∇ v | β 2 ( τ ) ( e + | ∇ v | ) | ∇ v | β 2 ( τ ) − 2 ∇ v = h 2 ( u , v )   in  D , u = 0 , v = 0   on  ∂ D . $$\begin{cases}-\mathrm{d}\mathrm{i}\mathrm{v}\left(\vert \nabla u{\vert }^{{\alpha }_{1}\left(\tau \right)-2}\nabla u+\mu \left(\tau \right)\left[\mathrm{log}\left(e+\vert \nabla u\vert \right)+\frac{\vert \nabla u\vert }{{\beta }_{1}\left(\tau \right) \left(e+\vert \nabla u\vert \right)}\right]\vert \nabla u{\vert }^{{\beta }_{1}\left(\tau \right)-2}\nabla u\right)={h}_{1}\left(u,v\right)\hfill & \quad \text{in\,}\mathfrak{D},\hfill \\ -\mathrm{d}\mathrm{i}\mathrm{v}\left(\vert \nabla v{\vert }^{{\alpha }_{2}\left(\tau \right)-2}\nabla v+\mu \left(\tau \right)\left[\mathrm{log}\left(e+\vert \nabla v\vert \right)+\frac{\vert \nabla v\vert }{{\beta }_{2}\left(\tau \right) \left(e+\vert \nabla v\vert \right)}\right]\vert \nabla v{\vert }^{{\beta }_{2}\left(\tau \right)-2}\nabla v\right)={h}_{2}\left(u,v\right)\hfill & \quad \text{in\,}\mathfrak{D},\hfill \\ u=0,\quad v=0\hfill & \quad \text{on\,}\partial \mathfrak{D}.\hfill \end{cases}$$ Here, D ⊂ R N $\mathfrak{D}\subset {\mathbb{R}}^{N}$ is a bounded domain with smooth boundary. By employing a variational principle due to B. Ricceri, together with critical point theory in the framework of Musielak–Orlicz–Sobolev spaces with logarithmic perturbations, we establish the existence of infinitely many weak solutions to the system.