American Institute of Mathematical Sciences

$ L_$φ$ u$:$=\sum $φ^ij$u_{ij}=0$

must be linear in $2D$. The function φ is assumed to have the Monge-Ampere measure $\det D^2 $φ bounded away from $0$ and $\infty$.

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$
$ u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L)) $

where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L _* >0 such that for L $\leq $L _* , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L _* , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L _* < L _** such that the least energy solution is trivial when L $\leq$L _*, the least energy solution is nontrivial when L $\in$(L _*,L _**], and the least energy solution does not exist when L >L _**. A connection with Delaunay surfaces in CMC theory is also made.

$\Delta_g v+\rho(\frac{h^* e^v}{\int_M h^* vd\mu(x)}-1)= 4\pi\sum_{j=1}^N\alpha_j(\delta_{q_i}-1)\quad\text{in }M,$(0.1)

where $(M,g)$ is a Riemann surface with its area $|M|=1$; or

$\Delta v+\rho\frac{h^*e^v}{\int_\Omega h^* e^vdx}=4\pi\sum_{j=1}^N\alpha_j \delta_{q_j}\quad\text{in }\Omega, $ (0.2)

where $\Omega$ is a bounded smooth domain in $ R^2$. Here, $\rho, \alpha_j$ are positive constants, $\delta_q$ is the Dirac measure at $q$, and both $h^*$'s are positive smooth functions. In this paper, we prove a sharp estimate for a sequence of blowing up solutions $u_k$ to (0.1) or (0.2) with $\rho_k\rightarrow\rho^*. Among other things, we show that for equation (0.1),

$\rho_k-\rho_*=\sum_{j=1}^\tau d_j( \Delta \log h^*(p_j)+\rho_*-N^*-2K(p_j)+o(1) )e^{-\frac{\lambda_k}{1+\alpha_j}}, $ (0.3)

and for equation (0.2),

$ \rho_k-\rho_*=\sum_{j=1}^\tau d_j(\Delta \log h^*(p_j)+o(1))e^{-\frac{\lambda_k}{1+\alpha_j}},$ (0.4)

where $\lambda_k\rightarrow+\infty$ and $d_j$ is a constant depending on $p_j$, a blow up point of $u_k$. See section 1 for more precise description. These estimates play an important role when the degree counting formulas are derived. The subsequent paper [19] will complete the proof of computing the degree counting formula.