Do you know that what is the equilibrium measure on [-1,1] if the kernel is $|x-y|^s$?
That is, for negative $s$ value I want to minimize the energy integral $$\int_{[-1,1]}\int_{[-1,1]}|x-y|^sf(x)f(y)dxdy,$$ for positive $s$ I am maximizing it. What is the optimal $f$ density function on $[-1,1]$? (I know the answer for the log kernel only.)
For $s\leq-1$, the energy equals $\infty$ for all densities on $[-1,1]$. For $-1<s\leq0$, the optimal density is given by $$f(x)=\frac{A}{(1-x^2)^{(1+s)/2}},$$ where $A$ is such that the measure has mass 1. For a reference, see p.163 of
N. S. LANDKOF, Foundations of Modern Potential Theory, Springer-Verlag, Heidelberg, 1972.