题目：On Gaussian Curvature Equation with Nonpositive Curvature

报告人：周风 教授 （华东师范大学 偏微分方程中心）

地点：致远楼101室

时间：2019年6月1日 星期六 下午 13:30-14:30

报告摘要：

We present some results concerning the solutions of $$ /Delta u +K(x) e^{2u}=0 /quad{/rm in}/;/; /mathbb{R}^2

$$ with $K/le 0$. We introduce the following quantity: $$/alpha_p(K)=/sup/left/{/alpha /in /R:/, /int_{/R^2} |K(x)|^p(1+|x|)^{2/alpha p+2(p-1)} dx<+/infty/right/}, /quad /forall/; p /ge 1.$$ Under the assumption $({/mathbb H}_1)$: $/alpha_p(K)> -/infty$ for some $p>1$ and $/alpha_1(K) > 0$, we show that for any $0 < /alpha < /alpha_1(K)$, there is a unique solution $u_/alpha$ with $u_/alpha(x) = /alpha /ln |x|+ c_/alpha+o/big(|x|^{-/frac{2/beta}{1+2/beta}} /big)$ at infinity and $/beta/in (0,/,/alpha_1(K)-/alpha)$.Furthermore, we show an example $K_0 /leq 0$ such that $/alpha_p(K_0) = -/infty$ for any $p>1$ and $/alpha_1(K_0) > 0$, for which we prove the existence of a solution $u_*$ such that $u_* -/alpha_*/ln|x| = O(1)$ at infinity for some $/alpha_* > 0$, but does not converge to a constant at infinity. The example exhibits also a new phenomenon of solutions with logarithmic growth and non-uniform bounded reminder term at infinity. This is a joint work with H.Y.Chen and D.Ye.

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