On Neumann superlinear elliptic problems

Abstract

In this paper we are going to show the existence of a nontrivial solution to the following model problem, \begin{equation*} \left\{\begin{array}{lll} -Δ(u) = 2uln(1+u^2)+\frac{|u|^2}{1+u^2}2u+u(sin(u)-cos(u)) \mbox{a.e. on } Ω\frac{\partial u}{\partial η} = 0 {a.e. on} \partial Ω. \end{array} \right. \end{equation*} 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.