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

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Abstract

For $N\geq 3$, by the seminal paper of Brezis and V\'eron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-\Delta u+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $1<q<N/(N-2)$ the existence and profiles near zero of all positive $C^1(\mathbb R^N\setminus \{0\})$ solutions are given by Friedman and V\'eron (Arch. Rational Mech. Anal. 96(4):359--387, 1986). In this paper, for every $q>1$ and $\theta\in \mathbb R$, we prove that the nonlinear elliptic problem (*) $-\Delta u-\lambda \,|x|^{-2}\,u+|x|^{\theta}u^q=0$ in $\mathbb R^N\setminus \{0\}$ with $u>0$ has a $C^1(\mathbb R^N\setminus \{0\})$ solution if and only if $\lambda>\lambda^*$, where $\lambda^*=\Theta(N-2-\Theta) $ with $\Theta=(\theta+2)/(q-1)$. We show that (a) if $\lambda>(N-2)^2/4$, then $U_0(x)=(\lambda-\lambda^*)^{1/(q-1)}|x|^{-\Theta}$ is the only solution of (*) and (b) if $\lambda^*<\lambda\leq (N-2)^2/4$, then all solutions of (*) are radially symmetric and their total set is $U_0\cup \{U_{\gamma,q,\lambda}:\ \gamma\in (0,\infty) \}$. We give the precise behavior of $ U_{\gamma,q,\lambda}$ near zero and at infinity, distinguishing between $1<q<q_{N,\theta}$ and $q>\max\{q_{N,\theta},1\}$, where $q_{N,\theta}=(N+2\theta+2)/(N-2)$. In addition, for $\theta\leq -2$ we settle the structure of the set of all positive solutions of (*) in $\Omega\setminus \{0\}$, subject to $u|_{\partial\Omega}=0$, where $\Omega$ is a smooth bounded domain containing zero, complementing the works of C\^{\i}rstea (Mem. Amer. Math. Soc. 227, 2014) and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).