About solutions for gradient-type cooperative systems beyond extremal parameter

Abstract

We investigate the existence and non-existence of solutions for cooperative elliptic gradient-type systems, depending on the real parameters λ and μ. Our approach, based on a refined analysis of the Nehari manifold associated with the problem, allows us to establish the existence and multiplicity of solutions by minimizing the associated energy functional over components of the Nehari set for parameters beyond the extremal parameter ${\lambda }^{⁎}\left(\mu \right)$.