A New Minimax Theorem for a Class of $C^1$-Functionals
We establish a new abstract minimax theorem for obtaining ground-state critical points of a class of \(C^{1}\)-functionals. The central feature of this class is that the associated Nehari set is not necessarily a differentiable manifold, but is homeomorphic to a (possibly non-complete) submanifold of the unit sphere. This occurs for functionals whose fibering maps \(t \mapsto I(tu)\) possess a unique global maximum point only when \(u\) belongs to an open cone in the underlying Banach space. A lower semicontinuous boundary barrier, which may be finite, controls minimizing sequences near the boundary of the spherical cone. As a consequence, we provide a general minimax characterization of the ground-state energy level. Elliptic applications complement existing results and include superlinear and quasilinear problems with finite barriers, including thresholds depending on the profile of a first eigenfunction and its gradient.