DEGENERATE BOUNDARY-VALUE PROBLEMS WITH A PERTURBING MATRIX FOR A DERIVATIVE

In the paper, there is considered degenerated Noether boundary value problem with a perturbing matrix for a derivative, in which the boundary condition is given by a linear vector functional. We have proposed an algorithm to consrtuct a set of linearly independent solutions of boundary value problems with a small parameter in the general case, when the number of boundary conditions given by a linear vector functional does not match with the number of unknowns in a degenerate differential system. There is used the technique of pseudoinverse Moore-Penrose matrices. Applying the Vishik-Lyusternik method, the solution of the boundary value problem is obtained as part of the Laurent series in powers of small parameter. We obtain conditions for the bifurcation of solutions of linear degenerated Noether boundary-value problems with a small parameter under the assumption that the unperturbed degenerated differential system can be reduced to central canonical form.

On the bifurcation of solutions of the Yamabe problem in product manifolds with minimal boundary

In this paper, we study the multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.

Spectral flow and bifurcation for a class of strongly indefinite elliptic systems

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can be obtained from the coefficients of the systems without using explicit solutions of their linearizations at the given branch. Our constructions are based on a comparison principle for the spectral flow and a generalization of a bifurcation theorem due to Szulkin.