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Multiple solutions to a perturbed Neumann problem. (English) Zbl 1387.35154
Summary: We consider the perturbed Neumann problem $\begin{cases} -{\Delta} u + \alpha(x)u=\alpha(x)f(u)+\lambda g(x,u)& \text{a.e.}\text{in } \Omega,\cr {\partial u}/{\partial \nu}=0&\text{on } \partial {\Omega},\end{cases}$ where $${\Omega}$$ is an open bounded set in $${\mathbb R}^N$$ with boundary of class $$C^2$$, $$\alpha\in L^\infty({\Omega})$$ with $$\text{ess\,inf}_{\Omega} \alpha >0$$, $$f:{\mathbb R}\rightarrow{\mathbb R}$$ is a continuous function and $$g:{\Omega}\times {\mathbb R}\rightarrow {\mathbb R}$$, besides being a Carathéodory function, is such that, for some $$p>N$$, $$\sup_{|s|\leq t}|g(\cdot,s)| \in L^p({\Omega})$$ and $$g(\cdot,t)\in L^\infty({\Omega})$$ for all $$t\in {\mathbb R}$$. In this setting, supposing only that the set of global minima of the function $$\frac{1}{2}\xi^2-\int_0^\xi f(t)\,dt$$ has $$M\geq 2$$ bounded connected components, we prove that, for all $$\lambda\in {\mathbb R}$$ small enough, the above Neumann problem has at least $$M+{}$$integer part of $${M}/{2}$$ distinct strong solutions in $$W^{2,p}({\Omega})$$.

##### MSC:
 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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