2026/07/16 by Francesco Paolo Maiale
#math.AP
We study strongly indefinite nonlinear Dirac functionals in the lowest partial-wave channel with radial coefficients in \mathbb R3. The symmetric model has the exact power nonlinearity F0(x,ψ)=b(|x|)|ψ|p/p, where 2<p<3 and the radial profile b is bounded, continuous, and uniformly positive. Compactness in this finite-angular-mode spinor channel and radial approximation-number estimates give quadratic growth of the symmetric minimax levels. For a non-even localized perturbation of order 1<τ<p/2, we impose a semiconvexity condition whose negative curvature is strictly smaller than the spectral gap. Together with the negative quadratic part and the convex exact-power core, this makes every negative spectral fiber uniformly strongly concave. Maximizing along that fiber reduces the problem to a C1 path on the positive spectral space. The polynomial Chambers--Ghoussoub--Bolle deformation theorem then yields infinitely many high-energy critical points of the restricted partial-wave functional; under the channel-invariance hypothesis these are weak solutions of the full Dirac equation. Convex perturbing primitives are covered without a smallness restriction on their amplitude. The same reduction applies after an even Hermitian quadratic term is absorbed into a renormalized Dirac operator, provided the renormalized operator has a gap at zero and the remaining perturbation satisfies the corresponding semiconvexity bound.