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# Geometric Analysis and Partial Differential Equations Seminar

## June 12, 2013 @ 10:30 am - 11:30 pm

##### Speaker

Dr Sanjiban Santra (University of Sydney)

##### Title

On the perturbed \$Q\$-curvature problem on \$\mathbb{S}^N\$.

##### Abstract

Let \$g_0\$ denote the standard metric on \$\mathbb{S}^4\$ and
\$P_{g_0}=\De^2_{g_{0}}- 2 \De_{g_{0}}\$ denote the corresponding Panietz operator. In this talk, we discuss the following fourth order elliptic problem with exponential nonlinearity :
\$\$ P_{g_{0}} u + 6 = 2Q(x)e^{4u} \mbox{ 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 on \$Q\$ assumed so far by Malchiodi-Struwe and Wei-Xu. We also prove uniqueness and exact multiplicity results for this problem. I will also discuss the case when \$N\geq 5.\$

## Details

Date:
June 12, 2013
Time:
10:30 am - 11:30 pm
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## Organiser

School of Mathematics and Applied Statistics

## Venue

Building 39C Room 174
University of Wollongong
Wollongong, 2500
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