Effects of Supercritical Perturbations on Nonlinear Elliptic Equations Driven by the (p, q)-Laplacian

Perera and Sim [1] established, among other results, the absence of non-trivial weak solutions for a class of critical elliptic equations driven by the ( p , q )-Laplacian in the unit ball. This paper investigates the effects of a variable perturbation, satisfying specific growth and regularity conditions, on this class of nonlinear critical elliptic equations. We demonstrate that the introduction of a supercritical variable perturbation can lead to the existence of a positive solution. This result highlights the critical role of perturbation strength in overcoming inherent solution constraints. Extending the work of do Ó et al. [2], which addressed the Laplacian case, we refine the analytical framework to accommodate more general conditions on the variable perturbation. Our findings provide new insights into the interplay between perturbations and the existence of solutions in nonlinear elliptic equations involving the ( p , q )-Laplacian.