A sign-changing solution for an asymptotically linear Schrödinger equation

We will present some recent results on the existence of nodal solutions for a class of asymptotically linear elliptic equations, which includes the special model case \begin{equation}\label{model}
-\Delta u + \lambda u = \frac{u^3}{1 + su^2}\quad \mbox{in }\ \mathbb{R}^N,
\end{equation} for \(N\geq 3\) and \(\lambda >0\). We find sign-changing solutions for a class of radially symmetric asymptotically linear Schrödinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda's Theorem.

This is a work in collaboration with Olimpio H. Miyagaki (UFJF, Brazil) and Sergio H. M. Soares (ICMC/USP, Brazil).