Patterns of Non-Radial Solutions to Coupled Semilinear Elliptic Systems on a Disc

Abstract

In this paper, we prove the existence of non-radial solutions to the problem $-△u=f(z,u)$, $u{|}_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb{C}:|z|<1\}$ with $u(z)\in {\mathbb{R}}^k$, where $f$ is a sub-linear continuous function, differentiable with respect to $u$ at zero and satisfying $f(e^{i\theta }z,u)=f(z,u)$ for all $\theta \in \mathbb{R}$, $f(z,-u)=-f(z,u)$. Under the assumption that $f$ respects additional (spacial) symmetries on ${\mathbb{R}}^k$, we investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example with extra $S_4$-symmetries.