Abstract. For p 12 11 , the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0, 1) are shown to form a Riesz basis of L 2 (0, 1) and a Schauder basis of L q (0, 1) whenever 1 < q < ∞.

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The validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p-Laplace operator p is investigated for p > 2. This problem is reduced to the comparison of the trivial solution (≡ 0, by hypothesis) with a nontrivial nonnegative solution u(x, t). The problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle. In this article, realistic counterexamples to the uniqueness of a nonnegative solution, the weak comparison principle, and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a Lipschitz nor an Osgood standard "uniqueness" condition. Nonnegative multi-bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub-and supersolutions that have supports of the same type.

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In this paper we give a generic necessary and sufficient condition for the solvability of2000 Mathematics Subject Classification: 34B15, 47H12.

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

This paper deals with existence results for the following nonlinear problem with the Dirichlet p-Laplacian D p in a bounded domain OCR N :NÞ is a fixed number, h hðx; sÞ is a given function from O Â R into R; and lAR stands for a spectral parameter. We focus on l close to l 1 ; including the resonant case l ¼ l 1 : The nonlinearity h is assumed to be of Landesman-Lazer type, but we can deal with vanishing nonlinearities as well. Our asymptotic method substitutes the Lyapunov-Schmidt method in some sense. Unlike in the semilinear case p ¼ 2; our method can treat more general nonlinearities if pa2 (vanishing nonlinearities with very fast decay).

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