A counterexample to the Fredholm alternative for the $p$-Laplacian
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- by Pavel Drábek and Peter Takáč PDF
- Proc. Amer. Math. Soc. 127 (1999), 1079-1087 Request permission
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 $p=2$).References
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Additional Information
- Pavel Drábek
- Email: pdrabek@kma.zcu.cz
- Peter Takáč
- Email: peter.takac@mathematik.uni-rostock.de
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1079-1087
- MSC (1991): Primary 34B15; Secondary 34C10
- DOI: https://doi.org/10.1090/S0002-9939-99-05195-3
- MathSciNet review: 1646309