Perturbation from symmetry for semiconvex lower-order partial-wave nonlinear Dirac equations

Abstract

We study strongly indefinite nonlinear Dirac functionals in the lowest partial-wave channel with radial coefficients in $\mathbb R^3$. The symmetric model has the exact power nonlinearity $F_0(x,\psi)=b(|x|)|\psi|^p/p$, 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 $1<\tau<p/2$, 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 $C^1$ 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.