**SMS scnews item created by Daniel Daners at Fri 7 Sep 2012 0900**

Type: Seminar

Modified: Fri 7 Sep 2012 1002

Distribution: World

Expiry: 10 Sep 2012

**Calendar1: 10 Sep 2012 1400-1500**

**CalLoc1: AGR Carslaw 829**

Auth: daners@bari.maths.usyd.edu.au

### PDE Seminar

# On a perturbed q-curvature problem in S4

### Santra

Sanjiban Santra

University of Sydney

10rd September 2012 2-3pm, AGR Carslaw 829

## Abstract

Let \(g_0\) denote the standard metric on \(\mathbb S ^4\)
and let \(P_{g_0}:=\Delta^2_{g_{0}}-2\Delta_{g_{0}}\) denote the
corresponding Paneitz operator. In this work, we study the fourth
order elliptic problem with exponential nonlinearity
\[ P_{g_{0}} u + 6 = 2Q(x)e^{4u}\]
on \(\mathbb{S}^4\). Here \(Q\) is a prescribed smooth function on
\(\mathbb{S}^4\) which is assumed to be a perturbation of a constant. We
prove existence results to the above problem under assumptions only on
the ``shape'' of \(Q\) near its critical points. These are more general
than the non-degeneracy conditions assumed so far. We also show local
uniqueness and exact multiplicity results for this problem. The main
tool used is the Lyapunov-Schmidt reduction.

Check also the PDE
Seminar page. Enquiries to Florica
Cîrstea or Daniel Daners.