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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 $W^{2,p}(\Omega )$.


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Additional Information

Biagio Ricceri
Affiliation: Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
Email: ricceri@dmi.unict.it

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08113-X
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.