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Abstract Multiple periodic solutions for the equation Δ(pn(Δxn−1)δ)+qnxnδ=∇F(n,xn),n∈Z, $$ \Delta \bigl(p_{n}(\Delta x_{n - 1})^{\delta } \bigr) + q_{n}x_{n}^{\delta } = \nabla F ( n,x_{n} ) , \quad n \in \mathbb{Z}, $$ are obtained via variational method and saddle-point theorem of Brezis and Nirenberg. Our main results extend some earlier results. An example is given.