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 {-Delta(p)u = lambda vertical bar u vertical bar(p-2)u + h(x, u(x); lambda) in Omega; u = 0 on partial derivative Omega, (P) Here, Omega is a bounded domain in R-N (N >= 1), Delta(p)u =(def) div(vertical bar del u vertical bar(p-2)del u) denotes the Dirichlet p-Laplacian on W-0(1,p) (Omega), 1 < p < infinity, and lambda is an element of R is a spectral parameter. Let mu(1) denote the. rst (smallest) eigenvalue of -Delta(p). Under some natural hypotheses on the perturbation function h : Omega x R x R -> R, we show that the trivial solution (0, mu(1)) is an element of E = W-0 (1,p) (Omega) x R is a bifurcation point for problem (P) and, moreover, there are two distinct continua, Z(mu 1)(+) and Z(mu 1)(-), consisting of nontrivial solutions (u, lambda) is an element of E to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, mu(1)). The continua Z(mu 1)(+) and Z(mu 1)(-) are either both unbounded in E, or else their intersection Z(mu 1)(+) boolean AND Z(mu 1)(-) contains also a point other than (0, mu(1)). 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(mu 1)(+) boolean AND Z(mu 1)(-) looks like (for p > 2) in an interesting particular case. Our proofs are based on very precise, local asymptotic analysis for. near mu(1) (for any 1 < p < infinity) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer's original work.