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 $0<\alpha \le 2$.

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