Positive and oscillatory radial solutions of semilinear elliptic equations

We prove that the nonlinear partial differential equation Δu+f(u)+g(|x|,u)=0, in ℝn,n≥3, with u(0)>0, where f and g are continuous, f(u)>0 and g(|x|,u)>0 for u>0, and limu→0+f(u)uq=B>0, for 1<q<n/(n−2), has no positive or eventually positive radial solutions. For g(|x|,u)≡0, when n/(n−2)≤q<(n+2)/(n−2) the same conclusion holds provided 2F(u)≥(1−2/n)uf(u), where F(u)=∫0uf(s)ds. We also discuss the behavior of the radial solutions for f(u)=u3+u5 and f(u)=u4+u5 in ℝ3 when g(|x|,u)≡0.