Multiplicity results for nonlinear Neumann boundary value problems involving p-Laplace type operators

Abstract

We consider the existence of at least two or three distinct weak solutions for the nonlinear elliptic equations $$ \textstyle\begin{cases} {-}\operatorname{div}(\varphi(x,\nabla u))+{|u|}^{p-2}u= \lambda f(x,u) &\mbox{in } \Omega, \varphi(x,\nabla u) \frac{\partial u}{\partial n}= \lambda g(x,u) & \mbox{on }\partial\Omega. \end{cases} $$ Here the function $\varphi(x,v)$ is of type $|v|^{p-2}v$ and the functions f, g satisfy a Caratheodory condition. To do this, we give some critical point theorems for continuously differentiable functions with the Cerami condition which are extensions of the recent results in Bonanno (Adv. Nonlinear Anal. 1:205-220, 2012) and Bonanno and Marano (Appl. Anal. 89:1-10, 2010) by applying Zhong’s Ekeland variational principle.