Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

((P))

Here, Ω is a bounded domain in denotes the Dirichlet p-Laplacian on , and is a spectral parameter. Let μ₁ denote the first (smallest) eigenvalue of −Δ _p . Under some natural hypotheses on the perturbation function , we show that the trivial solution is a bifurcation point for problem (P) and, moreover, there are two distinct continua, and , consisting of nontrivial solutions to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ₁). The continua and are either both unbounded in E, or else their intersection 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 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. Submitted: July 28, 2007. Accepted: November 8, 2007.