A borderline analysis of the Nehari manifold method for concave-convex system - Tripathi, Vinayak Mani; Silva, Kaye; Mishra, Pawan K. (Topological Methods in Nonlinear Analysis, 2025)
The aim of this paper is to obtain an existence and multiplicity result for a strongly coupled concave-convex system for an {\it optimal} choice of involved real parameters via the Nehari manifold method. In the paper, we have obtained the parametric region which is optimal in the sense that the constraint minimization idea based on the Nehari manifold is no longer applicable if the parameters lie in the exterior of the optimal region.By applying a finer analysis of fibering maps, we have shown the existence of atleast two positive solutions for the parameters lying below and even above the parametric optimal curve, characterized variationally via nonlinear generalized Rayleigh quotient.The main result of the paper is complemented by the study of the problem for negative parameter values.