Uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations

Chern, Jann-Long; Chen, Zhi-You; Tang, Yong-Li

doi:10.1090/S0002-9947-2011-05192-5

Uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations
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by Jann-Long Chern, Zhi-You Chen and Yong-Li Tang PDF

Trans. Amer. Math. Soc. 363 (2011), 3211-3231 Request permission

Abstract:

In this article, we are concerned with the semilinear elliptic equation \[ \Delta u+K(|x|)|u|^{p-1}u=0\quad \textrm {in}\ \mathbf {R}^n\setminus \{\mathbf {0}\},\] where $n>2$, $p>1$, and $K(|x|)>0$ in $\mathbf {R}^n$. The correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived. In addition, we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on $K$ and $p$. Furthermore, based on the Pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions, we obtain the whole structure of radial solutions depending on various situations.

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