  We consider Liouville-type and partial regularity results for the nonlinear
fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and
$n\ge1$. We give a complete classification of stable and finite Morse index
solutions (whether positive or sign changing), in the full exponent range. We
also compute an upper bound of the Hausdorff dimension of the singular set of
extremal solutions. Our approach is motivated by Fleming's tangent cone
analysis technique for minimal surfaces and Federer's dimension reduction
principle in partial regularity theory. A key tool is the monotonicity formula
for biharmonic equations.
