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.

中文翻译：

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$${\mathbb {R}}^3$$ 中单位球上半线性椭圆系统非径向解的存在性

在本文中，我们证明了单元上问题\(-\triangle u= f(x,u)\)、\(u|_{\partial \Omega }=0\)的非径向解的存在性球\(\Omega :=\{x\in {\mathbb {R}}^3: \Vert x\Vert <1\}\)与\(u(x)\in {\mathbb {R}}^ s\)，其中f是次线性连续函数，在零处相对于u可微，并且对于所有\(g\in O(3 ) 满足\(f(gx,u) = f(x,u)\) )\) , \( f(x,-u)=- f(x,u)\)。我们研究相应非径向解的对称性质。抽象结果由数值例子支持。