# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## A remark on positive radial solutions of the elliptic equation $\Delta u+K(\vert x\vert )u^ {(n+2)/(n-2)}=0$ in $\textbf {R}^ n$HTML articles powered by AMS MathViewer

by Yasuhiro Sasahara and Kazunaga Tanaka
Proc. Amer. Math. Soc. 123 (1995), 527-531 Request permission

## Abstract:

We consider the following semilinear elliptic equation involving critical Sobolev exponents: $\begin {array}{*{20}{c}} { - \Delta u = K(|x|){u^{(n + 2)/(n - 2)}}\quad {\text {in}}\;{{\mathbf {R}}^n},} \\ {u(x) \to 0\quad {\text {as}}\;|x| \to \infty ,} \\ \end {array}$ where $n \geq 3,K(r) \in C([0,\infty ),{\mathbf {R}})$. We prove the existence of a positive radial solution with asymptotic behavior $C/|x{|^{n - 2}}$ at $|x| = \infty$ under the conditions (i) $K(r) > 0$ for all $r > 0$, (ii) $K(0) = K(\infty )$, and (iii) there exist C, $\delta > 0$ such that $K(r) \geq K(0) - C{r^\delta }$ for small $r > 0$ and $K(r) \geq K(0) - C{r^{ - \delta }}$ for large $r > 0$.
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