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On the Fredholm Alternative for the $p$-Laplacian

Author(s): Paul A. Binding; Pavel Drábek; Yin Xi Huang
Journal: Proc. Amer. Math. Soc. 125 (1997), 3555-3559.
MSC (1991): Primary 35J65, 35P30
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Abstract: Consider

\begin{equation*}\left \{\begin{split} &-(|u'|^{p-2}u')'=\lambda |u|^{p-2}u+f(x), x\in (0, 1), &u(0)=\beta u'(0), \quad u'(1)=0,\end{split}\right . \end{equation*}

where $p>1$ and $\beta \in \mathbb{R}\cup \{\infty \}$ and let $\lambda _{1}$ be the principal eigenvalue of the problem with $f(x)\equiv 0$. For $\lambda =\lambda _{1}$, we discuss for which values of $p$ and $\beta $ the Fredholm alternative holds.

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Y.X. Huang and G. Metzen, The existence of solutions to a class of semilinear differential equations, Diff. Int. Equa. 8 (1995), 429-452. MR 95h:34034

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Additional Information:

Paul A. Binding
Affiliation: Department of Mathematics & Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Pavel Drábek
Affiliation: Department of Mathematics, University of West Bohemia, P.O. Box 314, 30614 Pilsen, Czech Republic

Yin Xi Huang
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: huangy@mathsci.msci.memphis.edu

DOI: 10.1090/S0002-9939-97-03992-0
PII: S 0002-9939(97)03992-0
Keywords: Fredholm alternative, the $p$-Laplacian
Received by editor(s): June 21, 1996
Additional Notes: Research of the authors was supported by NSERC of Canada and the I.W. Killam Foundation, the Grant # 201/94/0008 of the Grant Agency of the Czech Republic, and a University of Memphis Faculty Research Grant respectively
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1997, American Mathematical Society

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