A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Let (M,g,σ) be a compact Riemannian spin manifold of dimension m≥2, let S(M) denote the spinor bundle on M, and D Atiyah-Singer Dirac operator acting spinors ψ:M→S(M). We study existence solutions nonlinear equation with critical exponent(NLD)Dψ=λψ+f(|ψ|)ψ+|ψ|2m−1ψ where λ∈R f(|ψ|)ψ is subcritical nonlinearity in sense that f(s)=o(s2m−1) as s→∞. A model f(s)=αsp−2 20, even if λ an eigenvalue D. For some classes nonlinearities f also (NLD) λ∈R, except non-positive eigenvalues. If m≢3 (mod 4) finite number certain parameter ranges multiple (BND), near trivial branch, others away from it. The proofs are based variational methods using strongly indefinite functional associated to (NLD).