This paper focuses on the existence of positive solutions for the following weakly coupled Schrödinger system with supercritical growth except at the origin: $ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u_1 = \mu_1|u_{1}|^{p(r) - 2}u_1 + \beta|u_{2}|^{\frac{p(r)}{2}}|u_1|^{\frac{p(r)}{2}-2}u_{1}, & x\in B_1(0), \\ -\Delta u_2 = \mu_2|u_{2}|^{p(r) - 2}u_{2} + \beta|u_{1}|^{\frac{p(r)}{2}}|u_2|^{\frac{p(r)}{2}-2}u_{2}, & x\in B_1(0), \end{array} \right. \end{equation*} $ where $ B_1(0) $ is an unit ball $ {\mathbb{R}^N} $ with $ N\ge 3 ...
