Separation structure of positive radial solutions of a semilinear elliptic equation in \(\mathbf R^n\) (Q1414014)

Separation of positive radial solutions is studied for the elliptic equation \(\Delta u+ k(| x|)u^p= 0\), where \(\Delta= \sum^n_{i=1} {\partial^2\over\partial x^2_i}\), \(n\geq 3\), is the Laplace operator, \(p> 1\), and \(k\) is a continuous function in \(\mathbb R^n\setminus\{0\}\). If for some \(\ell> -2\) the function \(r^{-\ell}k(r)\) is decreasing to a positive constant as \(r\to\infty\), then the asymptotic behavior near \(\infty\) of the radial solutions is described in detail when \(n\) and \(p\) are sufficiently large. Using local separation of regular solutions, the author establishes the existence of singular solutions in case when every solution, with positive initial date at \(0\) exists globally. A new exponent, which is critical in the study of separation and intersection of singular solutions, is presented.