A Monotonicity Formula and a Liouville-type Theorem for a Fourth Order Supercritical Problem

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 $nge1$. 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 Flemings tangent cone analysis technique for minimal surfaces and Federers dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.