Anisotropic quasilinear elliptic systems with homogeneous critical nonlinearities

Abstract In this work, we consider a system of quasilinear elliptic equations driven by an anisotropic ‐Laplacian. The lower order nonlinearities are in potential form and exhibit critical Sobolev growth. We exhibit conditions on the coefficients of the differential operator, the domain of the unknown function, and the lower order nonlinearities under which nontrivial solutions are guaranteed to exist and conditions on these objects under which a nontrivial solution does not exist.