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A multiplicity theorem for the Neumann problem

Authors:	Biagio Ricceri

Institution:	Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

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

Keywords:	Neumann problem multiplicity of solutions global minima connected components

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