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$\left\{\begin{array}{cc}-\Delta u=\lambda u\hfill & \text{in}\Omega \hfill \\ u=0\hfill & \text{on}\partial \Omega \hfill \end{array}\right.$.

Classification:  35J60

@article{ASNSP_2009_5_8_3_429_0,
     author = {Grossi, Massimo},
     title = {On the shape of solutions of an asymptotically linear problem},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {3},
     year = {2009},
     pages = {429-449},
     zbl = {1182.35116},
     mrnumber = {2574338},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_3_429_0}
}

Grossi, Massimo. On the shape of solutions of an asymptotically linear problem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 3, pp. 429-449. http://www.numdam.org/item/ASNSP_2009_5_8_3_429_0/