Some multiplicity results for quasilinear Neumann problems

Abstract.

In this paper, using a recent two local minima result of B.Ricceri and the Palais-Smale property, we establish two results ensuring the existence of three solutions for the following Neumann problem:

$$ \left\{ {\begin{array}{*{20}l} {{ - \Delta _{p} u + \alpha {\left( x \right)}|u|^{{p - 2}} u = \alpha {\left( x \right)}f{\left( u \right)} + \lambda g{\left( {x,\,u} \right)}} \hfill} & {{{\text{in}}\;\Omega } \hfill} \\ {{\frac{{\partial u}} {{\partial v}} = 0} \hfill} & {{{\text{on}}\;\partial \Omega } \hfill} \\ \end{array} } \right.. $$