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 $\lambda ,\mu$, 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 $\gamma$ 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 $\gamma =1$, $\gamma >1$, and $\gamma <1$. The multiple results of multiplicity of solutions are based on a recent abstract critical point theorem for symmetric functionals on product spaces when $\gamma \ge 1$ and on a truncation argument for $\gamma <1$.