Three weak solutions for double phase elliptic problem with indefinite weight

This paper investigates the multiplicity of weak solutions for a double-phase elliptic problem with indefinite interaction, where the nonlinearity involves a potential term $ k(x) $ that may change sign and be singular within the domain. The problem is set up as a Dirichlet boundary value problem, where the differential operator has two different phases, involving two exponents $ p $ and $ q $ that meet a certain condition. Employing critical point theory, we demonstrate that at least one solution exists, and under suitable conditions, there are at least three solutions. These results come from applying abstract variational approaches, particularly the critical point theorems by Bonanno and Marano. The manuscript also presents a detailed variational framework and sets up the necessary preliminaries to support the main results.