Existence and nonexistence of sign-changing solutions for linearly perturbed superlinear equations on exterior domains

In this paper, we study radial solutions of $\mathrm{\Delta }u+K(\mathrm{\mid }x\mathrm{\mid })f(u)+\frac{(N-2{)}^{2}u}{\mathrm{\mid }x{\mathrm{\mid }}^{2+(N-2)\delta }}=0,0<\delta <2$ in the exterior of the ball of radius $R>0$ in ${\mathbb{R}}^N$ where $f$ grows superlinearly at infinity and is singular at $0$ with $f(u)\sim -\frac{1}{\mathrm{\mid }u{\mathrm{\mid }}^{q-1}u}$ and $0<q<1$ for small $u$. We assume $K(\mathrm{\mid }x\mathrm{\mid })\sim \mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ for large $\mathrm{\mid }x\mathrm{\mid }$ and establish the existence of an infinite number of sign-changing solutions when $N+q(N-2)<\alpha <2(N-1)\mathrm{.}$ We also prove nonexistence for $0<\alpha \le 2$.