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

TL;DR

This paper proves the existence of infinitely many radial solutions to a singular superlinear p-Laplacian equation on exterior domains, with solutions vanishing at infinity and specific growth conditions.

Contribution

It establishes the existence of infinitely many solutions for a singular superlinear p-Laplacian problem on exterior domains, extending previous results to singular and superlinear cases.

Findings

Infinitely many radial solutions exist.

Solutions tend to zero at infinity.

Conditions on parameters ensure solution existence.

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