Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity

Abstract

We consider the semilinear elliptic equation \begin{document} $Δ u + K(|x|)e^u = 0$\end{document} in \begin{document} $\mathbf{R}^N$\end{document} for \begin{document} $N > 2$\end{document} , and investigate separation phenomena of radial solutions. In terms of intersection and separation, we classify the solution structures and establish characterizations of the structures. These observations lead to sufficient conditions for partial separation. For \begin{document} $N = 10+4\ell$\end{document} with \begin{document} $\ell>-2$\end{document} , the equation changes its nature drastically according to the sign of the derivative of \begin{document} $r^{-\ell}K(r)$\end{document} when \begin{document} $r^{-\ell}K(r)$\end{document} is monotonic in \begin{document} $r$\end{document} and \begin{document} $r^{-\ell} K(r)\to1$\end{document} as \begin{document} $r\to∞$\end{document} .