Existence of non-radial solutions to semilinear elliptic systems on a unit ball in $${\mathbb {R}}^3$$

In this paper, we prove the existence of non-radial solutions to the problem \(-\triangle u= f(x,u)\), \(u|_{\partial \Omega }=0\) on the unit ball \(\Omega :=\{x\in {\mathbb {R}}^3: \Vert x\Vert <1\}\) with \(u(x)\in {\mathbb {R}}^s\), where f is a sub-linear continuous function, differentiable with respect to u at zero and satisfying \(f(gx,u) = f(x,u)\) for all \(g\in O(3)\), \( f(x,-u)=- f(x,u)\). We investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example.