A monotonicity formula and a Liouville-typetheorem for a fourth order super critical problem

Abstract

We consider Liouville-type and partial regularity results for then online arfourth-order problem

Δ2u = |u|p−1u in Rn,

where p>1 and n≥ 1.We give a complete classification of stableand finite Morse index solutions (whether positive o rsign changing),in the full exponent range.We also compute an upper bound. of the Hausdorff dimension of the singular set of extremal solutions.Our approachis motivated by Fleming’ stangent 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.