Multiple solutions to the double phase problems involving concave-convex nonlinearities

This paper is concerned with several existence results of multiple solutions for Schrödinger-type problems involving the double phase operator for the case of a combined effect of concave-convex nonlinearities. The first one is to discuss that our problem has infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the given problem. To establish such multiplicity results, we employ the fountain theorem and the dual fountain theorem as the primary tools, respectively. In particular we give the existence result of small energy solutions on a new class of nonlinear term.