Ngô Quốc Anh

April 15, 2014

Locally H^1-bounded for the Palais-Smale sequences in the region when the Gaussian curvature candidate is negative

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 16:22

In 1993, Ding and Liu announced their result about the multiplicity of solutions to the prescribed Gaussian curvature on compact Riemannian 2-manifolds of negative Euler characteristic. Their interesting result then published in the journal Trans. Amer. Math. Soc. in 1995, see this link.

Roughly speaking, by starting with the prescribed Gaussian curvature equation, i.e.

\displaystyle -\Delta u + k = Ke^{2u}

when the Euler characteristic \chi(M) is negative, i.e.

\displaystyle 2\pi \chi (M) = \int_M k e^{2u}dv_g <0,

they perturbed K using

\displaystyle K_\lambda = K+\lambda

where \lambda is a real number and the candidate function K is assumed to be

\displaystyle \max_{x \in M} K(x)=0.

Then they were interested in solving the following PDE

\displaystyle -\Delta u + k = K_\lambda e^{2u}.

Their main result can be stated as follows:

Theorem (Ding-Liu). There exists a \lambda^\star > 0 such that

  • the PDE has a unique solution for \lambda \leqslant 0;
  • the PDE has at least two solutions if 0<\lambda<\lambda^\star; and
  • the PDE has at least one solution when \lambda = \lambda^\star.

As always, the first solution was found as local minimizes of the energy functional associated to the PDE, i.e.

\displaystyle I_\lambda (u) = \int_M \big( |\nabla u|^2 +2ku-K_\lambda e^{2u} \big) dv_g.

To find the second solution, they used the mountain pass lemma. As such, one of the key steps is to check whether or not the functional I_\lambda satisfies the Palais-Smale condition. Loosely speaking, if \{u_i\}_i is any sequence in H^1(M) such that

  • I_\lambda (u_i) \to c for some c \in \mathbb R; and
  • I'_\lambda (u_i) \to 0 in the dual space H^{-1}(M) of H^1(M), then

\{u_i\}_i admits a convergent subsequence. To achieve this goal, they proved that the sequence \{u_i^+\}_i is locally H^1-bounded in the open set

M_-=\{x \in M : K_\lambda (x)<0\}

where they denoted u_i^+ = \max \{ u_i, 0\}. However, there was a sign problem in their paper which is recently pointed out by Michael Struwe and his co-authors in their preprint entitled “Large” conformal metrics of prescribed Gauss curvature on surfaces of higher genus. To be exact, instead of fixing the mistake by Ding and Liu i.e. fixing \lambda, they consider a sequence of solution of the PDE when \lambda is changing. However, their technique also works for the Ding and Liu situation. And in this note, we show how to fix the error.

To do so, we let B_r(p) \subset M_- and fix a smooth cut-off function 0 \leqslant \psi \leqslant 1 supported in B_{r/2}(p) and with \psi \equiv 1 on B_{r/4}(p). Then let \eta = \psi^2. Using \eta^2 u_i^+ as a test function, it follows from I'_\lambda (u_i) \to 0 that

\displaystyle \int_{B_{r/2}(p)} \Big( \nabla u_i^+\cdot \nabla (\eta^2 u_i^+)+2k\eta^2 u_i^+-K_\lambda e^{2u_i^+} \eta^2 u_i^+ \Big) dv_g \leqslant C\|\eta^2 u_i^+\| \leqslant C \|\eta u_i^+\|.

Here and in sequel we use C to denote various constants depending only on \psi, \varepsilon. Note that

\displaystyle \nabla u_i^+\cdot \nabla (\eta^2 u_i^+) = |\nabla( \eta u_i^+)|^2 - |\nabla \eta|^2 (u_i^+)^2

and here is the mistake in the Ding and Liu paper because they used a plus sign on the right hand side. Then there exists some small \varepsilon >0 such that

K_\lambda \leqslant -\varepsilon

in B_{r/2}(p). Thanks to e^{2t} \geqslant t^3 for any t, therefore, we obtain the following estimate

\displaystyle \int_{B_{r/2}(p)} \Big( |\nabla (\eta u_i^+)|^2 + \varepsilon \eta^2 (u_i^+)^4 \Big) dv_g \leqslant \int\limits_{B_{r/2}(p)} \Big( |\nabla \eta|^2 (u_i^+)^2 - k\eta^2 u_i^+\Big) dv_g+ C \|\eta u_i^+\|.

Recalling that \eta = \psi^2 and using the Young inequality, we can bound

\begin{array}{lcl} \displaystyle |\nabla \eta {|^2}{(u_i^ + )^2} &=& \displaystyle 4|\nabla \psi {|^2}{(\psi u_i^ + )^2}\\&\leqslant& C{(\psi u_i^ + )^2}\\ &\leqslant & \displaystyle \frac{1}{2}\varepsilon {(\psi u_i^ + )^4} + C\\ &=& \displaystyle \frac{1}{2}\varepsilon {\eta ^2}{(u_i^ + )^4} + C.\end{array}

Hence,

\displaystyle \int_{B_{r/2} (p)} \Big( |\nabla (\eta u_i^ + )|^2 + \frac{1}{2}\varepsilon \eta ^2 (u_i^ + )^4 \Big) dv_g \leqslant C - \int_{B_{r/2}(p)} k{\eta ^2}u_i^ + d{v_g} + C\|\eta u_i^ +\|.

Since

\displaystyle (u_i^+)^4 > (u_i^+)^2-1,

it is easy to see that the following

\displaystyle \varepsilon\|\eta u_i^ +\|^2 \leqslant C\|\eta u_i^ +\|+C

holds. Hence \|\eta u_i^ +\|\leqslant C and consequently, \{u_i^+\}_i is H^1-bounded in B_{r/2}(p). The proof is complete.

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