A multiplicity theorem for the Neumann problem

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