© 2002 by London Mathematical Society
On the Fredholm Alternative for the p-Laplacian in One Dimension
Centro de Modelamiento Matemático and Departamento de Ingenieria Matemática FCFM, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile manasevi{at}dim.uchile.cl
Fachbereich Mathematik, Universität Rostock D-18055 Rostock, Germany peter.takac{at}mathematik.uni-rostock.de
Received 23 October 2000. Revision received 17 April 2001.
We investigate the existence of a weak solution u to the quasilinear two-point boundary value problem
We assume that 1 < p < 
p ¬ = 2, 0 < a < , and that f 
L1(0,a) is a given function. The number 
k stands for the k-th eigenvalue of the one-dimensional p-Laplacian. Let p 
p x/a) denote the eigenfunction associated with 1; then p(k p x/a) is the eigenfunction associated with k. We show the existence of solutions to (P) in the following cases.
(i) When k=1 and f satisfies the orthogonality condition
the set of solutions is bounded.
(ii) If k=1 and ft L1(0,a) is a continuous family parametrized by t [0,1], with
then there exists some t* [0,1] such that (P) has a solution for f = ft*. Moreover, an appropriate choice of t* yields a solution u with an arbitrarily large L1(0,a)-norm which means that such f cannot be orthogonal to pp x/a.
(iii) When k 
2 and f satisfies a set of orthogonality conditions to p(k p x/a) 
on the subintervals 
, again, the set of solutions is bounded.
is a continuous family satisfying either
or another related condition, then there exists some t* [0,1] such that (P) has a solution for f = ft*.
Prüfer's transformation plays the key role in our proofs. 2000 Mathematical Subject Classification: primary 34B16, 47J10; secondary 34L40, 47H30.
Key Words: non-linear eigenvalue problem • Fredholm alternative • quasilinear two-point Dirichlet problem • one-dimensional p-Laplacian • Prüfer's transformation