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A multiplicity theorem for the Neumann problem

Author: Biagio Ricceri
Journal: Proc. Amer. Math. Soc. 134 (2006), 1117-1124
MSC (2000): Primary 35J20, 35J65
Published electronically: August 29, 2005
MathSciNet review: 2196046
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Abstract: Here is a particular case of the main result of this paper: Let $\Omega \subset {\mathbb{R}}^{n}$ be a bounded domain, with a boundary of class $C^{2}$, and let $f, g : {\mathbb{R}}\to {\mathbb{R}}$ be two continuous functions, $\alpha \in L^{\infty }(\Omega )$, with $\hbox{\rm ess inf}_{\Omega }\alpha >0$, $\beta \in L^{p}(\Omega )$, with $p>n$. If

\begin{displaymath}\lim_{\vert\xi \vert\to +\infty }{\frac{f(\xi )}{{\xi }}}=0 \end{displaymath}

and if the set of all global minima of the function $\xi \to {\frac{{\xi^{2}}}{{2}}}-\int _{0}^{\xi }f(t)\,dt$ has at least $k\ge 2$ connected components, then, for each $\lambda >0$ small enough, the Neumann problem

\begin{displaymath}\begin{cases} -\Delta u=\alpha (x)(f(u)-u) +\lambda \beta (x)... ... u}{\partial \nu }}=0&\text{on $\partial \Omega $ } \end{cases}\end{displaymath}

admits at least $k+1$ strong solutions in $W^{2,p}(\Omega )$.

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

  • 1. A. Ambrosetti, A perturbation theorem for superlinear boundary value problems, M. R. C. Tech. Summ. Rep. (1974).
  • 2. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • 3. Shujie Li and Zhaoli Liu, Perturbations from symmetric elliptic boundary value problems, J. Differential Equations 185 (2002), no. 1, 271–280. MR 1935639,
  • 4. Zhaoli Liu and Jiabao Su, Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 617–634. MR 2018870,
  • 5. Biagio Ricceri, Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. Nonlinear Convex Anal. 5 (2004), no. 2, 157–168. MR 2083908
  • 6. C. G. Simader, Higher regularity of weak $L_{q}$-solutions of the Neumann problem for the Laplacian, Bayreuth. Math. Schr., to appear.
  • 7. Guido Stampacchia, Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellitiche, Ann. Scuola Norm. Sup. Pisa (3) 12 (1958), 223–245 (Italian). MR 0125313
  • 8. Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. MR 768749

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Additional Information

Biagio Ricceri
Affiliation: Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Keywords: Neumann problem, multiplicity of solutions, global minima, connected components
Received by editor(s): June 10, 2004
Received by editor(s) in revised form: November 2, 2004
Published electronically: August 29, 2005
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.