The fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F99%3A00039168" target="_blank" >RIV/49777513:23520/99:00039168 - isvavai.cz</a>
Result on the web
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DOI - Digital Object Identifier
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Result language
angličtina
Original language name
The fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian
Original language description
In this work we study the range of the operator umapsto (|u'|^{p-2}u')'+lambda_1|u|^{p-2}u, u(0)=u(T)=0,p>1. We prove that all functions hinC^1[0,T] satisfying int^T_0 h(t)sin_p(pi_pt/T)dt=0 lie in the range, but that if pneq2 and hequiv0 the soltion set is bounded. Here sin(pi_pt/T)is a first eigenfunction associated to lambda_1. We also show that in this case the associated energy functional umapsto(1/p) int^T_0|u'|^p-(lambda_1/p) int^T_0|u|^p+int^T_0hu is unbounded from below if 1<p<2nd bounded from below (with a global minimizer) if p>2 on W^{1,p}_0 (0,T)(lambda_1 corresponds precisely to the best constant in the L^p-Poincaré inequality). Moreover, we show that unlike the linear case p=2, for pneq2 the range contains a nonempty opn set in L^{infty}(0,T).
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA201%2F97%2F0395" target="_blank" >GA201/97/0395: Topological and variational methods for nonlinear boundary value problems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>Z - Vyzkumny zamer (s odkazem do CEZ)

Publication year
1999
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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