The Fredholm Alternative for the p-Laplacian: Bifurcation from Infinity, Existence and Multiplicity (original) (raw)

This work is concerned with the existence and mul-tiplicity of weak solutions u ∈ W 1,p 0 (Ω) to the quasilinear elliptic problem (P)    −∆ p u = λ|u| p−2 u + f (x) in Ω ; u = 0 o n ∂ Ω , with the spectral parameter λ ∈ R near the (simple) principal eigenvalue λ 1 of the positive Dirichlet p-Laplacian −∆ p in a bounded domain Ω ⊂ R N , for 1 < p < ∞. Here, ∆ p u ≡ div(||u| p−2 u) and f ∈ L ∞ (Ω) is a given function. A pri-ori bounds on the solutions are obtained from a rather precise description of possible " large solutions " investigated by bifurca-tions from infinity. They take the form u = t −1 (ϕ 1 + v) as t → 0, t ∈ R \ {0}, where ϕ 1 stands for the (positive) eigenfunc-tion associated with λ 1 , and v is a relatively small perturbation of ϕ 1 which is orthogonal to ϕ 1. We also allow λ and f to vary with t → 0. Our method is based on the linearization of ∆ p near ϕ 1. As a result of our asymptotic formula for λ depending on t and f , with the integral Ω f ϕ 1 dx playing a major role, we are able to obtain a number of new results for problem (P). Some of these results for p ≠ 2 are quite different from the linear case p = 2.