Comparison theorems for radial solutions of semilinear elliptic equations (Q1117396)

Radially symmetric solutions to the elliptic equation \[ (E1)\quad \Delta u+f(u)=0 \] satisfy the ordinary differential equation \[ (E2)\quad u_{rr}+((n-1)/r)u_ r+f(u)=0,\quad u_ r(0)=0, \] where \(r=| x|\) and \(x\in {\mathbb{R}}^ n\). Let \(f(u)=u[1+g(u)]\), where g is even, positive and monotonically increasing. The author obtains a priori bounds for solutions of (E2) and studies the location of zeros as well as behavior of solutions for large r. The results are applied to the eigenvalue problem \[ \Delta w+\lambda w(1+| w|^{p-1})=0,\quad x\in \Omega;\quad w=0\quad on\quad \partial \Omega. \]