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

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 $0<\alpha \leq2$.