Let (M, g, sigma) be a compact Riemannian spin manifold of dimension m >= 2, let S(M) denote the spinor bundle on M, and let Dbe the Atiyah-Singer Dirac operator acting on spinors psi : M -> S(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent D psi = lambda psi + f(vertical bar psi vertical bar)psi + vertical bar psi vertical bar(2/m-1) psi (NLD) where lambda is an element of R and f(vertical bar psi vertical bar)psi is a subcritical nonlinearity in the sense that f(s) = o(s(2/m-1)) as s -> infinity. A model nonlinearity is f(s) = alpha s(p-2) with 2 0, even if lambda is an eigenvalue of D. For some classes of nonlinearities fwe also obtain solutions of (NLD) for every lambda is an element of R, except for non-positive eigenvalues. If m not equivalent to 3(mod 4) we obtain solutions of (BND) for every lambda is an element of R, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it. The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD). (C) 2021 Elsevier Inc. All rights reserved.
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