Entropy solutions for double-phase elliptic systems with $L^{1}$ source terms

We investigate nonlinear elliptic equations that describe double-phase phenomena with a reaction term that is independent of the gradient (i.e., it only satisfies the $L^{1}$-data condition), subject to homogeneous Dirichlet boundary conditions. Using the theory of Musielak–Orlicz–Sobolev spaces, and under fairly general conditions, we demonstrate the existence of entropy solutions. This is achieved through a regularization method combined with a priori estimates developed by Boccardo and Gallouët.