## Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces.(English)Zbl 1040.53046

The authors claim that the Leray-Schauder degree $$d(\rho)$$ of the nonlinear elliptic equation $\Delta_0 u+\rho\Biggl({h(x)e^u\over h(x) e^ud\mu}- 1\Biggr)= 0,\tag{1}$ on a compact Riemann surface $$(M,ds)$$, where $$\Delta_0$$ is the Beltrami-Laplace operator, is $$1$$ for $$0< \rho< 8\pi$$ and ${1\over m!} (-\chi(M)+ 1)\cdots(-\chi(M)+ m)\quad\text{for }8m\pi< \rho< 8(m+ 1)\pi,$ where $$\chi(M)$$ is the Euler characteristic class of $$M$$ [C. C. Chen and C. S. Lin, ibid. 56, 1667–1727 (2003; Zbl 1032.58010)]. To prove the claim, the estimate $\rho_k- 8m\pi= {2\over m} \sum^m_{j=1} h-1(p_{k,j})(\Delta_0 \log h(p_{k,j})+ 8m\pi- 2K(p_{k,j}))\lambda_{k,j} e^{-\lambda_{k,j}}+ O(e^{-\lambda_{k,j}}),\tag{2}$ where $$\lim_{k\to\infty}\rho_k= 8m\pi$$, $$p_{k,j}$$ are centers of the bubbles of $$u_k$$ and $$\lambda_{k,j}$$ are the local maxima of $$u_k$$, is proved in this paper (Theorem 1.1).
Equation (1) is the Euler-Lagrange equation of the nonlinear functional $J_\phi(\phi)= {1\over 2}\int_M| \nabla\phi|^2d\mu- \rho\log\Biggl(\int_M he^\phi d\mu\Biggr),$ and arises as the Nirenberg or Kazdan-Warner problem in geometry, from self-dual condensate solutions of some Chern-Simons-Higgs model in physics, and so on.
In Section 2, previous studies on (1) are reviewed. The main results in this section are cited from [Y. Y. Li, Commun. Math. Phys. 200, 421–444 (1999; Zbl 0928.35057)]. Then, estimating solutions $$u_k$$ for $$\rho_k$$ and \begin{aligned} \eta_{k,j}(x) &= u_k(x)- v_{k,j}(x)- (G^*_j(x)- G^*_j (p_{k,j})),\\ v_{k,j}(x) &= \log\Biggl({e^{\lambda_{k,j}}\over (1+ (\rho_k h_k(p_{k,j})/8) e^{\lambda_{k,j}}| x-q_{k,j}|^2)^2}\Biggr),\end{aligned} where $$\lambda_{k,j}= u_k(p_{k,j})$$, $$\nabla v_{k,j}(p_{k,j})= \nabla\log\widehat h(p_{k,j})$$ and $$\lambda_{k,j}= u_k(p_{k,j})= \max_{\overline B_{\delta_0(p)}} u_k(x)\to +\infty$$, $$G^*_j(x)= \rho_{k,j}(\widetilde G_j(x)+ \sum_{i\neq j}\rho_{k,j}G(x, p_{k,j})$$, $$\widetilde G_j(x)= G(x,p_{k,j})+ {1\over 2\pi}\log| x|$$, where $$G(x,p)$$ is the Green function of $$\Delta_0$$ with singularity at $$p$$, (2) is proved in Section 3. Some estimates are proved in Section 4. A sharper estimate of $$\eta_{k,j}(x)$$ used in the calculation of the Leray-Schauder degree is given in Section 5 (Theorem 5.1). In Section 6 similar estimates for the Dirichlet problem of (1) on $$\mathbb{R}^2$$ are given (Theorem 6.1–6.3).

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35J60 Nonlinear elliptic equations 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 58E12 Variational problems concerning minimal surfaces (problems in two independent variables)

### Citations:

Zbl 1032.58010; Zbl 0928.35057
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