I have been trying hard to solve these equations. There are four equations in total:
$$ \begin{align*} px^{p-1} + qx^{q-1} \lambda &= 0 \\ py^{p-1} + qy^{q-1} \lambda &= 0 \\ pz^{p-1} + qz^{q-1} \lambda &= 0 \\ x^q + y^q + z^q &= 1 \end{align*} $$ Here $0<p<q$ and $x,y,z>0$; $\lambda $ is a constant.