Existence of infinitelymany solutions for double phase problem with sign-changing potential

In this paper, we investigate the existence of infinitely many solutions for the following double phase problem {- div(|. u| p- 2. u + a(x)|. u| q- 2. u) = f (x, u), in Omega, u = 0, on partial derivative Omega where N = 2 and 1 < p < q < N. Based on a direct sum decomposition of a space W1, H 0 (Omega), we prove that the above problem possesses multiple solutions under mild assumptions on a and f. The primitive of the nonlinearity f is of super- q growth near infinity in u and allowed to be sign- changing. Furthermore, our assumptions are suitable and different from those studied previously.