I have to solve the following scalar non-linear equation. \begin{equation*} \xi^{\frac{2}{\varepsilon}}+(\xi-k)^{\frac{2}{\varepsilon}}=1 \end{equation*}
with respect to $\xi\geq 0$. Here $\varepsilon>0$ and $k\in[0,1]$ are given parameters. So far I'm using the bisection method, and it works fine. However, I would like to know if it is possible to get a closed form solution in function of $\varepsilon,k$. This is possible in the simple case $k=0$, indeed we have \begin{equation*} 2\,\xi^{\frac{2}{\varepsilon}}=1 \end{equation*} and so \begin{equation*} \xi =\left(\frac{1}{2}\right)^{\frac{\varepsilon}{2}} \end{equation*} but what can we say if $k\neq0$? Another simple case is the one where $\varepsilon=2$, so that we have \begin{equation*} \xi+(\xi-k)=1 \end{equation*} and so \begin{equation*} \xi = \frac{1+k}{2} \end{equation*}