Calculus of Variations and Geometric Measure Theory

Monotonicity and symmetry of solutions with uniform limits for some elliptic systems

Nicola Soave (Università degli Studi di Torino)

created by malchiodi on 11 Oct 2017

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

Scuola Normale Superiore, Aula Tonelli


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 $\Lambda > 0$,


$ -\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.