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We construct a smooth radial positive solution for the following m -coupled elliptic system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_i= f(u_i)-\beta \sum \limits _{j\ne i} u_i u_j^2,&{}\quad \hbox {in}\quad B_1(0),\\ u_i=0, i=1,\ldots ,m,&{}\quad \hbox {on}\quad \partial B_1(0), \end{array}\right. \end{aligned}$$ for \(\beta >0\) large enough, where \(f\in C^{2,1}({\mathbb {R}}),\ f(0)=0\) , \(B_1(0)\subset \mathbb R^N\) is the unit ball centered at the origin, \(m\ge 3,\ N\ge 1\) are positive integers. Our main result is an extension of Casteras and Sourdis (J Funct Anal 279:108674, 2020) from \(m=2\) to general case \(m\ge 3\) under some natural and essential non-degeneracy conditions by gluing method. The way we construct is somehow different and greatly simplify the computations since we overcome the difficulties brought by too much parameters from multiple equations.
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