30 oct 2017 -- 15:00 [open in google calendar]

Scuola Normale Superiore, Aula Tonelli

**Abstract.**

I will present some results regarding sharp a priori bounds, Liouville-type theorems, monotonicity and $1$-dimensional symmetry for solutions of the two systems

$ -\Delta u =u-u^3-\Lambda uv^2$ in $\mathbb{R}^N;$ $ -\Delta v =v-v^3-\Lambda u^2v$ in $\mathbb{R}^N$, $u,v \ge 0$ in $\mathbb{R}^N$, with $\Lambda > 0$,

and

$ -\Delta u \, =-uv^2$ in $\mathbb{R}^N;$ $ -\Delta v \, =-u^2v$ in $\mathbb{R}^N$, $u,v \ge 0$ in $\mathbb{R}^N$,

under suitable assumptions on the behavior of $u$ and $v$ when $x_N \to \pm \infty$. The talk is based on joint works with Alberto Farina, Berardino Sciunzi and Susanna Terracini.