Abstract

In this paper, we aim at investigating a class of nonlinear elliptic problems involving double-phase operators with variable exponents. More precisely, we consider differential operators of the form
{div(\left|\xi\right|}^{p\left(x\right)-2}\xi+\mu\left(x\right)\left|\xi\right|^{q\left(x\right)-2}\xi)\ 
together with a nonlocal term. Such problems arise in a variety of applications, including reaction–diffusion processes, nonlinear elasticity, non-Newtonian fluid mechanics, and quantum physics.

One of the distinguishing features of the double-phase operator is its ability to continuously switch between different elliptic phases depending on the spatial position. This phenomenon significantly increases the analytical complexity of the associated problems and distinguishes them from classical (p)-Laplacian models. The reaction term considered in this work combines both subcritical and critical growth nonlinearities, thereby encompassing a broader class of equations than those treated in many previous studies.

Our main objective is to establish the existence of weak solutions and to analyze their qualitative properties. The problem is studied in an appropriate Musielak–Orlicz–Sobolev space associated with the double-phase structure. The variational approach serves as the primary analytical framework, allowing the solutions to be characterized as critical points of the corresponding energy functional. To this end, we verify the geometric conditions required by critical point theory and investigate the compactness properties of the functional. While compactness follows directly from Sobolev-type embeddings in the  subcritical case, the presence of critical growth leads to a loss of compactness. To overcome this difficulty, we employ a concentration–compactness principle of Lions type adapted to the present setting, which enables us to recover the compactness necessary for the variational analysis. Our results contribute and complement several existing results and provide a flexbile variational framework that can be applied to classes with similar differential operators.