Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms

Abstract

In this paper we investigate existence and multiplicity of solutions for a critical growth anisotropic quasilinear elliptic system, depending on two real parameters $λ,\,μ$, that is coupled through a subcritical perturbation term and involves a negative critical part. We identify a specific scaling for the system and introduce a parameter $γ$ associated with this scaling, which governs the geometry of the corresponding variational functional. We establish three distinct types of multiplicity results, depending on the regimes $γ= 1$, $γ> 1$, and $γ< 1$. The multiple results of multiplicity of solutions are based on a recent abstract critical point theorem for symmetric functionals on product spaces when $γ\geq1$ and on a truncation argument for $γ<1$.