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Conformal metrics on $${R}^n$$ with arbitrary total $$Q$$-curvature

Dr. Ali Hyder,Department of Mathematics and Informatics University of Basel, Switzerland
 Speaker Dr. Ali Hyder,Department of Mathematics and Informatics University of Basel, Switzerland Jul 13, 2017 from 04:00 PM to 05:00 PM LH 006 vCal iCal

Abstract: I will talk about the existence of solution to the $$Q$$-curvature problem

$$(1)$$

$$(-\Delta)$$$$^\frac n2$$ u=$$Qe$$$$^{nu}$$$$\quad\text{in }$$$$\mathbb{R}^n,$$$$\quad \kappa:$$=$$\int_{\mathbb{R}^n}$$$$Qe^{nu}dx$$$$<\infty,$$

where $$Q$$ is a non-negative function and $$n>2$$. Geometrically, if $$u$$ is a solution to $$(1)$$ then $$Q$$ is the $$Q$$-curvature of the conformal metric $$g_u$$ = $$e^{2u}$$$$|dx|^2$$ $$(|dx|^2$$ is the Euclidean metric on $$\mathbb{R}^n)$$, and $$\kappa$$ is the total $$Q$$-curvature of $$g_u$$.

Under certain assumptions on $$Q$$ around origin and at infinity, we prove the existence of solution to $$(1)$$for every $$\kappa$$ > $$0$$.

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