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

Abstract

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