AI 中文总结
研究\(\mathbb{R}^3\)中最低部分波通道的强不定非线性狄拉克泛函，通过施加半凸性条件等方法，利用多项式变形定理得出受限部分波泛函的多个高能临界点，这些点在通道不变性下是完整狄拉克方程的弱解，且凸扰动无幅度小性限制。
AI 中文摘要
我们研究了\(\mathbb{R}^3\)中具有径向系数的最低部分波通道中的强不定非线性狄拉克泛函。对称模型具有精确的幂次非线性\(F_0(x,\psi)=b(|x|)|\psi|^p/p\)，其中\(2 < p < 3\)且径向分布\(b\)有界、连续且一致为正。在这个有限角模旋量通道中的紧致性和径向近似数估计给出了对称极小极大水平的二次增长。对于阶数为\(1 < \tau < p/2\)的非偶局部扰动，我们施加一个半凸性条件，其负曲率严格小于谱隙。这使得每个负谱纤维均匀地强烈凹。沿该纤维最大化将问题简化为正谱空间上的\(C^1\)路径。多项式钱伯斯 - 古苏布 - 博勒变形定理然后产生受限部分波泛函的无限多个高能临界点；在通道不变性假设下，这些是完整狄拉克方程的弱解。凸扰动原函数在其幅度上没有小性限制。在一个偶厄米二次项被吸收到重整化狄拉克算子后，同样的简化也适用，前提是重整化算子在零处有一个间隙并且其余扰动满足相应的半凸性界。
英文摘要
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 $F_0(x,ψ)=b(|x|)|ψ|^p/p$, 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 $C^1$ 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.
