Infinitely many positive solutions for a double phase problems involving the double phase operator

In this paper we study a double phase problem which involves the double phase operator, and the nonlinear term has an oscillatory behavior. By using variational methods and the theory of the Musielak-Orlicz-Sobolev space, we establish the existence of infinitely many solutions whose $W_0^{1,H}(\Omega)$-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively).