On the shape of solutions of an asymptotically linear problem

In this paper we study the problem $(0.1)\left\{\begin{array}{cc}-\Delta u={\left|u\right|}^{ϵ}u\hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{on}\partial \Omega \hfill \end{array}\right.$where $\Omega$ is a smooth bounded domain of ${ℝ}^N$, $N\ge 1$, $ϵ>0$. We will show that, under some assumptions, the solutions to (0.1) are close to suitable linear combinations of eigenfunctions of the problem$\left\{\begin{array}{cc}-\Delta u=\lambda u\hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{on}\partial \Omega \hfill \end{array}\right.$.