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Geometric Analysis and Partial Differential Equations Seminar

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

Geometric Analysis and Partial Differential Equations Seminar Series at UOW
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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Venue

Building 39C Room 174
University of Wollongong
Wollongong, 2500
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