This paper exhibits a study of a class of strongly singular nonlinear integral equations concerning the uniqueness of the solution (sufficient conditions) and the nonpositiveness of the corresponding eigenvalue problem. The particular operatorial form of the studied equations emerges from the crack problems in a homogeneous linear elastic body. There is an important feature concerning the contribution of this article: the uniformly valid asymptotic solutions of the strongly singular integral equations specific to crack analysis. Finally, some perturbation method is analyzed and the results are illustrated treated by means of an example based on the solution of W. T. Koiter [Quart. J. Mech. Appl. Math. 53, 164-178 (1955; Zbl 0064.191)] and other actual problems.