A generalization of Morse lemma and its applications

The problem concerning harmonic mappings is addressed by using the Morse splitting lemma. As a first step, a generalization of Morse splitting lemma to the non-Hilbertian case is derived. Following the development of the generalization, a result on critical groups in different topologies similar to the results about semilinear elliptic equation and harmonic mapping is established. The results obtained are presented as Theorem 2.5, Corollaries 2.7 and 2.8. As applications, the semilinear elliptic equation, harmonic mappings, and the equation of constant mean curvature are presented.