Existence and nonexistence of sign-changing solutions for linearly perturbed superlinear equations on exterior domains

TL;DR

This paper investigates the existence and nonexistence of sign-changing solutions for a class of superlinear elliptic equations with singular and perturbation terms on exterior domains, establishing conditions for infinitely many solutions or none.

Contribution

It provides new criteria for the existence of infinitely many sign-changing solutions in perturbed superlinear equations on exterior domains, extending previous results to singular and linearly perturbed cases.

Findings

Infinite sign-changing solutions exist when N+q(N-2) < α < 2(N-1)

Nonexistence of solutions for 0<α ≤ 2

Conditions depend on the growth rate of K(|x|) and the singularity of f(u)

Abstract

In this paper, we study radial solutions of $\Delta u + K(|x|) f(u)+\frac{ (N-2)^2 u}{|x|^{2+(N-2)\delta}} =0, \ 0<\delta<2$ in the exterior of the ball of radius $R>0$ in ${\mathbb R}^{N}$ where $f$ grows superlinearly at infinity and is singular at $0$ with $f(u) \sim -\frac{1}{|u|^{q-1}u}$ and $0<q<1$ for small $u$. We assume $K(|x|) \sim |x|^{-\alpha}$ for large $|x|$ and establish the existence of an infinite number of sign-changing solutions when $N+q(N-2) <\alpha <2(N-1).$ We also prove nonexistence for $0<\alpha \leq2$.