Three solutions for a Neumann problem

Abstract

In this paper we consider a Neumann problem of the type $$ \cases -\Delta u = \alpha (x) (\vert u\vert^{q-2}u-u)+\lambda f(x,u) &\text{in } \Omega, \\ \displaystyle {\partial u\over \partial \nu}=0 &\text{on } \partial\Omega. \endcases \tag \hbox{$\text{\rm P}_{\lambda}$} $$ Applying the theory developed in [< i> On a three critical points theorem< /i> , Arch. Math. (Basel) < b> 75< /b> (2000), 220–226], we establish, under suitable assumptions, the existence of an open interval $\Lambda\subseteq \mathbb R$ and of a positive real number $\varrho$, such that, for each $\lambda\in\Lambda$, problem (P$_{\lambda}$) admits at least three weak solutions in $W^{1,2}(\Omega)$ whose norms are less than $\varrho$.