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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A counterexample to the Fredholm alternative
for the $\lowercase{p}$-Laplacian


Authors: Pavel Drábek and Peter Takác
Journal: Proc. Amer. Math. Soc. 127 (1999), 1079-1087
MSC (1991): Primary 34B15; Secondary 34C10
MathSciNet review: 1646309
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Abstract | References | Similar Articles | Additional Information

Abstract: The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional $p$-Laplacian with $1 < p < \infty$ is considered:

\begin{equation*}- (|u'|^{p-2} u')' - \lambda |u|^{p-2} u = f(x) \;\mbox{ for } 0 < x < T ; \quad u(0) = u(T) = 0 , \tag{*}\end{equation*}

where $f\equiv 1 + h$ with $h\in L^\infty (0,T)$ small enough. Solvability properties of Problem (*) with respect to the spectral parameter $\lambda\in \mathbb{R}$ are investigated. We focus our attention on some fundamental differences between the cases $p\neq 2$ and $p=2$. For $p\neq 2$ we give a counterexample to the classical Fredholm alternative (which is valid for the linear case $p=2$).


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Additional Information

Pavel Drábek
Email: pdrabek@kma.zcu.cz

Peter Takác
Email: peter.takac@mathematik.uni-rostock.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05195-3
PII: S 0002-9939(99)05195-3
Keywords: Nonuniqueness and multiplicity of solutions, resonance for the $p$-Laplacian, nonlinear Fredholm alternative
Received by editor(s): July 16, 1997
Additional Notes: The first author’s research was supported in part by the Grant Agency of the Czech Republic, Project 201/97/0395, by the Ministry of Education of the Czech Republic, Project Nr.\ VR 97156, and by the University of Rostock, Germany.
The second author’s research was supported in part by Deutsche Forschungsgemeinschaft (Germany).
Communicated by: Jeffrey Rauch
Article copyright: © Copyright 1999 American Mathematical Society