A low-energy multiwell genus principle below a concentration–compactness threshold

We establish a windowed symmetric minimax principle for even C 1 functionals on real, separable, reflexive Banach spaces when Palais–Smale compactness is available at levels in ( 0 , Λ ⁎ ) . Using the Krasnosel'skii genus together with Benci's pseudo-index, we localize the complete minimax–deformation mechanism to the proper energy window 0 < J < Λ ⁎ . Every admissible deformation fixes the complement of this window, every minimax level lies below Λ ⁎ , and a plateau of length r forces genus at least r for the corresponding critical set. A single m -dimensional low-energy well therefore yields at least m nontrivial antipodal critical pairs; nested wells give the infinite-dimensional version. For a fractional p -Laplacian problem with critical growth, concentration–compactness identifies the exact compactness threshold and localized test spaces yield canonical well levels. Sharpness is established by an Ekeland sequence on the critical Nehari manifold: global Palais–Smale compactness fails at the threshold for an admissible perturbation, whereas an explicit parameter inequality places any prescribed finite number of minimax levels strictly below it.