Asymptotic bifurcation problems for quasilinear equations existence and multiplicity results

Abstract

In this paper we address the existence and multiplicity results for´− ∆pu − λ|u| p−2 u = h(x, u) in Ω, u = 0 on ∂Ω, where p > 1, ∆pu = div(|∇u| p−2 ∇u), h is a bounded function and the spectral parameter λ stays "near" the principal eigenvalue of the p-Laplacian. We show how the bifurcation theory combined with certain asymptotic estimates yield desired results.

Multiplicity results and global bifurcations for a degenerate elliptic equation

“…The system E p λ (δ) can also be put in an integral equation form: (9) and (10) which, in turn, are equivalent to one of the following integral equations …”

“…Even there has been a significant amount of work that has been done on degenerate elliptic equations, and in particular, bifurcation problems for pLaplacian equations, not only for one-dimensional problems but in higher dimensions as well, in the last 25 years. Authors who have contributed to this field include, for instance, Laurent Veron, M.Guedda [16], Pavel Drabek [3], [8], [9], [10], [11], [12], Bryan Rynne [4], [5], [13], [14], [15], and many others. However, until recently, the basic spectral properties obtained in the case p > 2 and the exact numbers of solutions of problem E p λ and especially the existence of collection of intervals I n , n = 1, 2, 3 .…”