Critical sets of 2-dimensional compact manifolds.
∗Partially supported by Grant MM409/94 Of the Ministy of Science and Education, Bulgaria. ∗∗Partially supported by Grant MM442/94 of the Ministy of Science and Education, Bulgaria.Let M be a complete C1−Finsler manifold without boundary and f : M → R be a locally Lipschitz function. The classical proof of the well known deformation lemma can not be extended in this case because integral lines may not exist. In this paper we establish existence of deformations generalizing the classical result. This...
We consider the nonlinear eigenvalue problem in with . A condition on indefinite weight function is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in . A nonexistence result is also given for the case .
We consider the Yamabe type family of problems , in , on , where is an annulus-shaped domain of , , which becomes thinner as . We show that for every solution , the energy as well as the Morse index tend to infinity as . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...
Equivalence and zero sets of certain maps on infinite dimensional spaces are studied using an approach similar to the deformation lemma from the singularity theory.