On a class of critical Schrödinger--Poisson systems involving the $(p,q)$-Laplacian

This paper investigates a class of Schrödinger-Poisson systems in $\mathbb{R}^3$ featuring the $(p,q)$-Laplacian operator and a combination of critical and subcritical nonlinearities in the Schrödinger equation while the $m$-Laplacian and a power type nonlinearity in the Poisson’s one. We consider both the attractive and repulsive cases, which correspond to different signs in front of the nonlocal term. While most existing literature relies on auxiliary functionals or specialized techniques to overcome the lack of compactness and ensure the boundedness of Palais-Smale sequences, we employ a direct variational approach. By applying the Mountain Pass Theorem and concentration compactness principles, we establish the existence of positive solutions. A careful analysis is conducted to identify the parameter ranges for which the Mountain Pass level falls within the compactness threshold, despite the technical challenges posed by the unbalanced growth of the operator and the nonlocal interaction.