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Can you please help me solve the following nonlinear equation?

\begin{equation} \boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{m}\right)^{\frac{m}{m-1}}\left(-\frac{m}{m-1}\right) \left(\boldsymbol{1}^{\top}\left(-\alpha\varepsilon \ln (\boldsymbol{z})-\boldsymbol{\nu}\right)^{m}\right)^{\frac{2-m}{m-1}} \left(-\alpha\varepsilon \ln (\boldsymbol{z})-\boldsymbol{\nu}\right)^{m-1}.\label{33b} \end{equation}

All the parameters ($\boldsymbol{\Gamma}$, $y$, $\beta$, $m$, $\boldsymbol{\nu}$, $\alpha$, and $\varepsilon$) are given, and I want to solve this equation for $\boldsymbol{z}$.

For the case that $m=2$, I can use the Lambert W function and solve it but I am wondering if there is a general solution for general $m$.

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    $\begingroup$ What is $\odot$? $\endgroup$
    – LSpice
    Commented Aug 11, 2022 at 3:52
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    $\begingroup$ The symbol $\odot$ denotes the elementwise multiplication. $\endgroup$
    – Iman Nodozi
    Commented Aug 11, 2022 at 6:32

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