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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
DOI: https://doi.org/10.1090/S0002-9939-99-05195-3
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 $p=2$).


References [Enhancements On Off] (What's this?)

  • [1] A. Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 725-728. MR 89e:35124
  • [2] P. A. Binding, P. Drábek, Y. X. Huang, On the Fredholm alternative for the $p$-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 3555-3559. MR 98b:35058
  • [3] P. A. Binding, P. Drábek, Y. X. Huang, On the range of the $p$-Laplacian, Appl. Math. Letters 10 (1997), 77-82. MR 98g:34035
  • [4] E. A. Coddington and N. Levinson, ``Theory of Ordinary Differential Equations'', McGraw-Hill, Inc., New York, 1955. MR 16:1022b
  • [5] K. Deimling, ``Nonlinear Functional Analysis'', Springer-Verlag, Berlin-Heidelberg-New York, 1985. MR 86j:47001
  • [6] J. I. Díaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 521-524. MR 89e:35051
  • [7] J. Fleckinger, J. Hernández, P. Taká\v{c} and F. de Thélin, Uniqueness and positivity for solutions of equations with the $p$-Laplacian, Proceedings of the Conference on Reaction-Diffusion Equations, Trieste, Italy, October 1995. Marcel Dekker, Inc., New York-Basel, 1997. MR 98h:35074
  • [8] S. Fu\v{c}ík, J. Ne\v{c}as, J. Sou\v{c}ek and V. Sou\v{c}ek, ``Spectral Analysis of Nonlinear Operators'', Lecture Notes in Mathematics, Vol. 346, Springer-Verlag, New York-Berlin-Heidelberg, 1973. MR 57:7280
  • [9] P. J. McKenna, W. Reichel and W. Walter, Symmetry and multiplicity for nonlinear elliptic differential equations with boundary blow-up, Nonlinear Anal. 28(7) (1997), 1213-1225. MR 97h:35049
  • [10] M. A. del Pino, M. Elgueta and R. F. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')' + f(t,u) = 0$, $u(0) = u(T) = 0$, $p>1$, J. Differential Equations 80(1) (1989), 1-13. MR 91i:34018
  • [11] M. A. del Pino and R. F. Manásevich, Multiple solutions for the $p$-Laplacian under global nonresonance, Proc. Amer. Math. Soc. 112(1) (1991), 131-138. MR 91h:34026
  • [12] M. A. del Pino, R. F. Manásevich and A. E. Murúa, Existence and multiplicity of solutions with prescribed period for a second order quasilinear o.d.e., Nonlinear Anal. 18(1) (1992), 79-92. MR 92m:34055
  • [13] W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian, J. Inequalities and Appl. 1 (1997), 47-71.

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Additional Information

Pavel Drábek
Email: pdrabek@kma.zcu.cz

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

DOI: https://doi.org/10.1090/S0002-9939-99-05195-3
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

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