# Ngô Quốc Anh

## March 12, 2014

### H^1 boundedness of a sequence of solutions to the prescribing Gaussian curvature problem in the negative case

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 22:31

In a previous note, we showed how to prove an $H^1$ boundedness of a sequence of solutions $\{u_k\}_k$ to the following PDE

$\displaystyle -\Delta u_k +\alpha_k = R(x)e^{u_k}$

in the negative case, i.e. $\alpha _k\searrow \alpha$ for some $\alpha <0$ as $k \to \infty$. In this note, we do not change $\alpha_0$, instead, we are going to change $R$. More precisely, we are interested in some $H^1$ boundedness of a sequence of solutions $\{u_k\}_k$ to the following PDE

$\displaystyle -\Delta u_k +\alpha = R_k(x)e^{u_k}$

for some $R_k = R+\lambda_k$ with $\lambda_k \searrow 0$ as $k \to \infty$. This note is adapted from a recent preprint by Michael Struwe et al., see this.

As always, we assume that $\alpha <0$ is constant and $(M,g)$ is a closed, connected Riemannian surface with smooth background metric $g$. We further assume for the sake of simplicity that $\text{vol}(M,g)=1$.

Step 1. We claim for sufficiently large $k$ that

$\displaystyle \int_M R_k^4 e^{u_k} \leqslant C(R)$

for some constant $C$ depending only on $\|R\|_{C^1}$ and on $(M,g)$.

Proof of Step 1. To prove this, we observe that

$\displaystyle \int_M R_k =\int_M e^{-u_k} \big( -\Delta u_k +\alpha\big) = -\int_M |\nabla u_k|^2 e^{-u_k} + \alpha\int_M e^{-u_k},$

which implies, by $\alpha <0$, for large $k$ that

$\displaystyle\int_M |\nabla u_k|^2 e^{-u_k}\leqslant \int_M R_k < 1+\int_M R.$