The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

Abstract

In this paper, we study and discuss the existence of multiple solutions of a class of non–linear elliptic equations with Neumann boundary condition, and obtain at least seven non–trivial solutions in which two are positive, two are negative and three are sign–changing. The study of problem (1.1): \( \left\{ {\begin{array}{*{20}l} {{ - \Delta u + \alpha u = f(u),} \hfill} & {{x \in \Omega } \hfill} \\ {{\frac{{\partial u}} {{\partial \gamma }} = 0,} \hfill} & {{x \in \partial \Omega ,} \hfill} \\ \end{array} } \right. \)is based on the variational methods and critical point theory. We form our conclusion by using the sub–sup solution method, Mountain Pass Theorem in order intervals, Leray–Schauder degree theory and the invariance of decreasing flow.