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

In this paper, we prove the existence of non-radial solutions to the problem $-\triangle u=f(z,u)$, $u|_{\partial D}=0$ 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.