On the classification of radial solutions for \(\Delta u+K(| x|)e^{2u}=0\) in \(\mathbb{R}^ n\) (Q1329276)

This paper mostly concerns the existence and asymptotic behavior at \(\infty\) of solutions \(u: \mathbb{R}_ +\to \mathbb{R}\) of the differential equation (1) \(u''+ {n-1\over x} u'+ f(x) e^{2u}= 0\) for \(n\geq 3\) and a nontrivial locally Hölder continuous function \(f\) in \(\mathbb{R}_ +\) satisfying various additional technical conditions. First existence and nonexistence theorems are proved. Then sufficient conditions on \(f\) are given for every solution \(u(x)\) to have various asymptotic behavior at \(\infty\), including \(C+ o(1)\), \(-C\log x+ O(1)\), and \(-{1\over 2} \log(\log x)+ O(1)\) as \(x\to\infty\) for a positive constant \(C\). Specific asymptotic formulas are obtained if \(f(x)\) is a power function.