Multiple solutions of conformal metrics with negative total curvature

Abstract

In this paper, we consider the Gaussian curvature equation \begin{equation}{\label{eq:01}} {\Delta} u+K(x)e^{2u}=0 \quad \mbox{in} \ {{\bf R}}^2 , \end{equation} where ${\Delta}=\sum_{i=1}^{2}\frac{\partial^2}{\partial x_i^2}$ is the Laplace operator in ${{\bf R}}^2$. $K(x)$ is always assumed to be of one sign for $|x|$ large. For each $K$, we introduce $\alpha_1=\alpha_1(K)$ by \begin{equation} \alpha_1=\sup \{\alpha \in {{\bf R}} : \int_{{{\bf R}}^2} |K(x)|(1+|x|^2)^{\alpha}\,dx <+{\infty} \} . \tag*{(0.1)} \end{equation} Suppose that $\alpha_1>0$, $K(x)$ is positive somewhere in ${{\bf R}}^2$ and satisfies \begin{equation} \int_{{{\bf R}}^2}K(x)\,dx <0 . \tag*{(0.2)} \end{equation} We prove that there exists $0 <\alpha_0\leq\alpha_1$ such that for any given $\alpha \in (0,\alpha_0)$, there exist at least two solutions of (0.1) with $-2\pi \alpha$ as their total curvature.