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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 $N\geq 5.$