Multiplicity Results and Global Bifurcations for a Degenerate Elliptic Equation

Copyright c © 2013 M. Amattat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the question of determining the exact number of so-lutions of the problem Epλ below We distinguish two cases: whether 1 < p 2 or 2 < p < ∞. In the first case, we shall show that the problem Epλ behaves like in the semi-linear case which corresponds to p = 2.. In the second case and under some conditions which will be specified later; we shall show that the spectrum for problem Epλ consists of a collection of intervals In, n = 1, 2, 3, whose ends points are mem-bers of the sequences (λn)n, (μn)n where the first one is the sequence of eigenvalues for the pseudo-Laplacian operator and the other one is a sequence of a kind of eigenvalues for Epλ. And each time λ = μn, there exists secondary bifurcating continuum εn of singular solutions which are diffeomorphic to [0, π]n, n = 1, 2.