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A multiplicity theorem for the Neumann problem. (English) Zbl 1134.35345
Summary: 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 $$\text{ess inf}_{\Omega}\alpha >0$$, $$\beta \in L^{p}(\Omega)$$, with $$p>n$$. If $\lim_{|\xi |\to +\infty}{\frac{f(\xi)}{{\xi}}}=0$ 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 $$k\geq 2$$ connected components, then, for each $$\lambda >0$$ small enough, the Neumann problem $\begin{cases} -\Delta u=\alpha(x)(f(u)-u) +\lambda \beta(x)g(u)&\text{ in } \Omega\\ \frac{\partial u}{\partial \nu}=0&\text{ on }\partial \Omega\end{cases}$ admits at least $$k+1$$ strong solutions in $$W^{2,p}(\Omega)$$.

##### MSC:
 35J60 Nonlinear elliptic equations 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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