Global structure of positive solutions to equations of Matukuma type (Q1815417)

The authors consider the semilinear elliptic equation \[ \Delta u+{1\over 1+|x|^\tau} u^p=0,\quad x\in\mathbb R^n, \] where \(n>2\), \(p>1\) and \(\tau>0\). Any positive radial solution of this equation satisfies \[ (r^{n-1}u_r(r))_r+{r^{n-1}\over 1+r^\tau} u^+(r)^p=0,\quad r>0,\quad u(0)=\alpha>0,\tag{*} \] where \(r=|x|\) and \(u^+=\max\{u,0\}\). The authors study the structure of solutions of \((*)\) in \((\alpha,p)\)-space, and, in particular, study the structure of the sets of slowly decaying solutions, rapidly decaying solutions, and crossing solutions.