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

35J60 Nonlinear elliptic equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
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