On the Fredholm Alternative for the p-Laplacian in One Dimension |
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Authors: | Manasevich Raul F; TakaC Peter |
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Institution: | 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 |
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Abstract: | We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L1(0,a) is a given function. The number k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Letp p x/a) denote the eigenfunction associated with 1; then p(kp x/a) is the eigenfunction associated with k. We show the existenceof 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 parametrizedby t 0,1], with then there exists some t* 0,1] such that (P) has a solutionfor f = ft*. Moreover, an appropriate choice of t* yields asolution u with an arbitrarily large L1(0,a)-norm which meansthat such f cannot be orthogonal to pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto 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. |
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Keywords: | non-linear eigenvalue problem Fredholm alternative quasilinear two-point Dirichlet problem one-dimensional p-Laplacian Prü fer's transformation |
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