On oscillating radial solutions for non-autonomous semilinear elliptic equations

We consider semilinear elliptic equations of the form $\mathrm{\Delta }u+f(|x|,u)=0$ on ${\mathbb{R}}^N$ with $f(|x|,u)=q(|x|)g(u)$. These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity $f$, there exists an oscillating radial solution converging to a zero of the function $g$. We also study the oscillating and limiting behavior of this solution.