In the last few decades, lots of work on first eigenfunction of $p$-Laplace with Dirichlet and other boundary conditions. But I couldn't find much on periodic boundary conditions. I have computed the first eigenfunction for $p=3$, square domain in $\Bbb R^2$. I have shown it as a grey scale image below. One period I tiled in the plane ($4\times 4$ tiles) to illustrate its meeting periodic boundary conditions.
I wonder such a simple structure not have a closed form expression. Can we guess anything on its closed form expression? Any reference to work in this direction (especially in dimension greater than $1$.)