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Abstract

We consider the perturbed Neumann problem $$ \cases{ -{\mit\Delta} u + \alpha(x)u=\alpha(x)f(u)+\lambda g(x,u)& \hbox{a.e. in ${\mit\Omega}$},\cr {\partial u}/{\partial \nu}=0&\hbox{on $\partial {\mit\Omega}$,}} $$ where ${\mit\Omega}$ is an open bounded set in $\mathbb{R}^N$ with boundary of class $C^2$, $\alpha\in L^\infty({\mit\Omega})$ with $\mathop{\rm ess\,inf}_{\mit\Omega} \alpha >0$, $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $g:{\mit\Omega}\times \mathbb{R}\rightarrow \mathbb{R}$, besides being a Carathéodory function, is such that, for some $p>N$, $\sup_{|s|\leq t}|g(\cdot,s)|\in L^p({\mit\Omega})$ and $g(\cdot,t)\in L^\infty({\mit\Omega})$ 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 $W^{2,p}({\mit\Omega})$.