Two Infinite Families of Solutions for Singular Superlinear Equations on Exterior Domains

TL;DR

This paper investigates radial solutions to a class of singular superlinear elliptic equations in exterior domains, establishing the existence of two infinite solution families under specific growth and singularity conditions.

Contribution

It introduces two infinite families of solutions for singular superlinear equations in exterior domains, expanding understanding of such equations with specific growth and singularity behaviors.

Findings

Existence of two infinite families of solutions.

Solutions exist under conditions on parameters N, q, and α.

Characterization of solution behavior at infinity and near singularity.

Abstract

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