Solutions of $x^{\alpha }+y^{\alpha }=1$ in $\lambda$-Cantor sets

In this paper, we study the structure of the solutions of $x^{\alpha }+y^{\alpha }=1$ with both coordinates in a $\lambda$-Cantor set. We prove that, for any $\alpha >1$, the intersection with the middle-third Cantor set has positive Hausdorff dimension, strengthening the infinitude question posed by Jiang et al. [J. London Math. Soc. 113 (2026), e70408]. To explore the structure of the solutions, for arbitrary $\alpha$, we find that their Hausdorff dimension undergoes phase transitions as $\lambda$ varies: we establish the occurrence of Hausdorff phase transitions by proving the existence of zero-dimensional and positive-dimensional regions.