Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems (original) (raw)

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem −Δpu = λ|u| p−2 u + h x, u(x); λ in Ω ; u = 0 on ∂Ω. (P) Here, Ω is a bounded domain in R N (N ≥ 1), Δpu def = div(|∇u| p−2 ∇u) denotes the Dirichlet p-Laplacian on W 1,p 0 (Ω), 1 < p < ∞, and λ ∈ R is a spectral parameter. Let μ1 denote the first (smallest) eigenvalue of −Δp. Under some natural hypotheses on the perturbation function h : Ω× R × R → R, we show that the trivial solution (0, μ1) ∈ E = W 1,p 0 (Ω) × R is a bifurcation point for problem (P) and, moreover, there are two distinct continua, Z + μ 1 and Z − μ 1 , consisting of nontrivial solutions (u, λ) ∈ E to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ1). The continua Z + μ 1 and Z − μ 1 are either both unbounded in E, or else their intersection Z + μ 1 ∩ Z − μ 1 contains also a point other than (0, μ1). For the semilinear problem (P) (i.e., for p = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union Z + μ 1 ∩ Z − μ 1 looks like (for p > 2) in an interesting particular case. Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer's original work.