Multiple solutions of the Ambrosetti–Rabinowitz problem

Abstract

In this paper, we consider the following elliptic problem $\left(\begin{array}{cc}-\Delta u=f(x,u),& \text{in}\Omega ,\\ u=0,& \text{on}\partial \Omega ,\end{array}\right)\left(P\right)$ where the nonlinearity $f$ satisfies the Ambrosetti–Rabinowitz condition. Using an additional growth condition of $f$ at a bounded region, we can obtain five nontrivial solutions of $\left(P\right)$ by applying homological linking arguments and Morse theory.