Multiplicity results and global bifurcations for a degenerate elliptic equation

Abstract

We consider the question of determining the exact number of solutions of the problem E p λ below We distinguish two cases: whether 1 < p 2 or 2 < p < ∞ . In the first case, we shall show that the problem E p λ 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 E p λ consists of a collection of intervals I n , n = 1, 2, 3, whose ends points are members of the sequences (λ n ) n , (μ n ) 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 E p λ . And each time λ = μ n , there exists secondary bifurcating continuum ε n of singular solutions which are diffeomorphic to [0, π] n , n = 1, 2..