Positive and oscillatory radial solutions of semilinear elliptic equations (Q1357900)

The authors consider the nonlinear partial differential equation \[ \Delta u+f(u)+g(|x|,u)=0 \tag{1} \] in \(\mathbb{R}^n\), \(n\geq 3\), where \(u(0)>0\), \(f\) and \(g\) are continuous, \(f(u)>0\) and \(g(|x|,u)>0\) for \(u>0\). Under some assumptions on \(f\) and \(g\) including mild conditions on the order of the growth of \(f(u)\) to zero as \(u\to 0\), they prove the nonexistence of positive or eventually positive radial solutions to the equation (1) with \(u(0)>0\).