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

TL;DR

This paper establishes Liouville-type and partial regularity results for a nonlinear fourth-order PDE in Euclidean space, classifying stable solutions and bounding the singular set's dimension using a monotonicity formula.

Contribution

It provides a complete classification of stable and finite Morse index solutions for the supercritical biharmonic equation, extending previous results to the full exponent range.

Findings

Classified all stable and finite Morse index solutions.

Derived an upper bound for the Hausdorff dimension of the singular set.

Developed a monotonicity formula for biharmonic equations.

Abstract

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.