Large blow-up sets for Q-curvature equations

Abstract:

On a bounded domain of the Euclidean space $\mathbb{R}^{2m}$, m>1, Adimurthi, Robert and Struwe pointed out that, even assuming a volume bound $\int e^{2mu} dx\le C$, some blow-up solutions for prescribed Q-curvature equations $(-\Delta)^m u= Q e^{2m u}$ without boundary conditions may blow-up not only at points, but also on the zero set of some nonpositive nontrivial polyharmonic function. This is in striking contrast with the two dimensional case (m=1). During this talk, starting from a work in progress with Ali Hyder and Luca Martinazzi, we will discuss the construction of such solutions which involve (possible generalizations of) the Walsh-Lebesgue theorem and some issues about elliptic problems with measure data.