Existence of solutions for double-phase nonlinear elliptic equations with singular potentials, nonlocal interactions, and mixed boundary conditions

This work examines multiplicity results for weak solutions to a double phase elliptic problem with mixed boundary conditions, a Hardy-type potential, and a nonlocal source term that may contain singularities within the domain $\mathcal{D} $. The paper establishes two primary results. The first utilizes critical point theory to demonstrate the existence of at least two distinct solutions in the superlinear case. The second relies on an alternative critical point theorem to show the existence of three weak solutions in the sublinear case. These results contribute to the development of nonlinear problem theory in mathematical physics, tackling the challenges related to the double phase structure, Hardy-type singularities, mixed boundary conditions, and nonlocal effects.