Ngô Quốc Anh

June 18, 2011

A note on the equation involving the prescribing Gaussian curvature problem, 2

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 17:28

Followed by this note, the trick use in that note also works for the following equation

\displaystyle -\Delta u +K_0(x)=K_1(x)e^{2u}+K_2(x)e^{-2u}, \quad x \in M.

In fact, by letting \overline u=2u, then our PDE becomes

\displaystyle -\Delta \overline u +2K_0(x)=2K_1(x)e^{\overline u}+2K_2(x)e^{-\overline u}.

Let v be a solution of the following PDE

\displaystyle -\Delta v=2K_0(x)-2\overline K_0

where \overline K_0 is the average of K_0 over M. We let w=\overline u+v. Then it is easy to verify that w solves the following

\displaystyle -\Delta w +2\overline K_0=2K_1(x)e^{-v}e^w+2K_2(x)e^ve^{-w}.

Finally, letting

\alpha=2\overline K_0, \quad R_1(x)=2K_1(x)e^{-v(x)}, \quad R_2(x)=2K_2(x)e^{v(x)}

we get that

\displaystyle -\Delta w +\alpha = R_1(x)e^w+R_2(x)e^{-w}.

Unfortunately, it is hard to see the relation between \alpha and \chi(M), the characteristic of M.

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