Existence of an Infinite Number of Solutions to a Singular Superlinear p-Laplacian Equation on Exterior Domains

In this paper, we prove the existence of an infinite number of radial solutions of the $p$-$Laplacian$ equation ${\mathrm{\Delta }}_{p}u+K(\mathrm{\mid }x\mathrm{\mid })f(u)=0$ on the exterior of the ball of radius $R>0$ in ${\mathbb{R}}^N$ such that $u(\mathrm{\mid }x\mathrm{\mid })\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ where $f$ grows superlinearly at infinity and is singular at $0$ with $f(u)\sim -\frac{1}{\mathrm{\mid }u{\mathrm{\mid }}^{m-1}u}$ and $0<m<1$ for small $u$. We also assume $K(\mathrm{\mid }x\mathrm{\mid })\sim \mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ for large $\mathrm{\mid }x\mathrm{\mid }$ where $N+\frac{m(N-p)}{p-1}<\alpha <2(N-1)\mathrm{.}$