Existence result for a Neumann problem

Abstract

In this paper we are going to show the existence of a nontrivial solution to the following model problem, $\{\begin{array}{l}-\Delta (u)=2uln(1+u^2)+\frac{|u{|}^2}{1+u^2}2u+usin(u)a.e.on\Omega \frac{\partial u}{\partial \eta }=0a.e.on\partial \Omega .\end{array}\}$ As one can see the right hand side is superlinear. But we can not use an Ambrosetti-Rabinowitz condition in order to obtain that the corresponding energy functional satisfies (PS) condition. However, it follows that the energy functional satisfies the Cerami (PS) condition.