A multiplicity theorem for the Neumann problem
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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 $\operatorname {ess\ inf}_{\Omega }\alpha >0$, $\beta \in L^{p}(\Omega )$, with $p>n$. If \begin{equation*} \lim _{|\xi |\to +\infty }{\frac {f(\xi )}{{\xi }}}=0 \end{equation*} 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{equation*} \begin {cases} -\Delta u=\alpha (x)(f(u)-u) +\lambda \beta (x)g(u)&\text {in $\Omega $}, \\ \dfrac {\partial u}{\partial \nu }=0 & \text {on $\partial \Omega $} \end{cases} \end{equation*} admits at least $k+1$ strong solutions in $W^{2,p}(\Omega )$.References
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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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1117-1124
- MSC (2000): Primary 35J20, 35J65
- DOI: https://doi.org/10.1090/S0002-9939-05-08113-X
- MathSciNet review: 2196046