On the Spectrum zero problem of nonlinear Dirac equations

The Spectrum Zero Problem for the nonlinear Dirac equation is treated. The main theorem proves to be the first result to address solutions of the nonlinear Dirac equation with spectrum point zero. Addressing this problem presents three significant challenges, which cannot be handled by existing methods. First, the spectrum of the Dirac operator is unbounded both from below and from above. Second, there is a need for embedding theorems in working space, and third, there is a lack of $L^2$-boundedness for sequences. These challenges are addressed by introducing new ingredients to prove the main theorem. Specifically, a new sequence is constructed by perturbing the functional, the uniform boundedness of this sequence is demonstrated, and then the compactness principle is applied to translate the sequence into the solution.