Sharp existence and classification results for nonlinear elliptic equations in $\mathbb R^N\setminus\{0\}$ with Hardy potential

Abstract

For $N\geq 3$, by the seminal paper of Brezis and Véron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-Δu+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $1 1$ and $θ\in \mathbb R$, we prove that the nonlinear elliptic problem (*) $-Δu-λ\,|x|^{-2}\,u+|x|^θu^q=0$ in $\mathbb R^N\setminus \{0\}$ with $u>0$ has a $C^1(\mathbb R^N\setminus \{0\})$ solution if and only if $λ>λ^*$, where $λ^*=Θ(N-2-Θ) $ with $Θ=(θ+2)/(q-1)$. We show that (a) if $λ>(N-2)^2/4$, then $U_0(x)=(λ-λ^*)^{1/(q-1)}|x|^{-Θ}$ is the only solution of (*) and (b) if $λ^*<λ\leq (N-2)^2/4$, then all solutions of (*) are radially symmetric and their total set is $U_0\cup \{U_{γ,q,λ}:\ γ\in (0,\infty) \}$. We give the precise behavior of $ U_{γ,q,λ}$ near zero and at infinity, distinguishing between $1 \max\{q_{N,θ},1\}$, where $q_{N,θ}=(N+2θ+2)/(N-2)$. In addition, for $θ\leq -2$ we settle the structure of the set of all positive solutions of (*) in $Ω\setminus \{0\}$, subject to $u|_{\partialΩ}=0$, where $Ω$ is a smooth bounded domain containing zero, complementing the works of C\^ırstea (Mem. Amer. Math. Soc. 227, 2014) and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).