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 $N+\frac{m(N-p)}{p-1}<\alpha <2(N-1).$

Submitted to a journal for publication. arXiv admin note: substantial text overlap with arXiv:2507.17863