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.

(more…)

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