On a perturbed q-curvature problem in S4

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 $P_{g_0}:={\mathrm{\Delta }}_{g_0}^2-2{\mathrm{\Delta }}_{g_0}$ 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.

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