Abstract : Let a,b functionals on a real Banach space X. We consider the nonlinear eigenvalue problem a'(u) = lambda b'(u), lambda an element of R, u an element of N sub alpha = (u an element of X:b(u) = alpha) for a fixed alpha. We allow that a', b' are indefinite and the level set N sub alpha is unbounded. We get finite and infinite lower bounds for the number of eigenvectors on N sub alpha depending on alpha. Furthermore, we study the weak convergence of the eigenvectors u against zero and the convergence of the eigenvalues lambda against zero. Our applications are concerned with eigenvalue problems for nonlinear elliptic partial differential equations where the principal elliptical part is indefinite, and with Hammerstein integral equations where the kernel has eigenvalues of different signs. The main abstract theorems in Section 5 provide a general formulation of the Ljusternik-Schnirelman theory in Banach spaces in the constrained case. (Author)