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 \(P_{g_0}:=\Delta^2_{g_{0}}-2\Delta_{g_{0}}\) 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.

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