Existence and nonexistence of weak solutions for a coupled $p, q$-Laplacian system with critical nonlinearities

In this paper, we study the existence and nonexistence of weak solutions for a coupled elliptic system involving the p -Laplacian and q -Laplacian operators. Under suitable assumptions on the exponents and the parameters, we establish the existence of a nontrivial weak solution by variational methods, combining the Mountain Pass Theorem with the Nehari manifold approach. We also investigate the critical coupling case and prove a nonexistence result for nontrivial nonnegative solutions on bounded smooth strictly star-shaped domains by means of a Pohozaev-type identity.