Abstract

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem Delta(2)u = vertical bar u vertical bar(p-1)u in R-n, where p > 1 and n >= 1. 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. (C) 2014 Elsevier Inc. All rights reserved.