No abstract

Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.

We consider resonance problems at an arbitrary eigenvalue of the p-Laplacian, and prove the existence of weak solutions assuming a standard Landesman Lazer condition. We use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem.

A version of Sturm--Liouville theory is given for the one-dimensional p-Laplacian including the radial case. The treatment is modern but follows the strategy of Elbert's early work. Topics include a Prüfer-type transformation, eigenvalue existence, asymptotics and variational principles, and eigenfunction oscillation.

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 < ∞.