Sharp existence and classification results for nonlinear elliptic equations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo>?</mml:mo><mml:mo stretchy="false">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:math> with Hardy potential

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 $-\Delta u+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $11$ $\theta\in \mathbb R$, we prove that nonlinear elliptic problem (*) u-\lambda \,|x|^{-2}\,u+|x|^{\theta}u^q=0$ with $u>0$ has a solution only $\lambda>\lambda^*$, where $\lambda^*=\Theta(N-2-\Theta) $ $\Theta=(\theta+2)/(q-1)$. We show (a) $\lambda>(N-2)^2/4$, then $U_0(x)=(\lambda-\lambda^*)^{1/(q-1)}|x|^{-\Theta}$ is (b) $\lambda^*<\lambda\leq (N-2)^2/4$, radially symmetric their total set $U_0\cup \{U_{\gamma,q,\lambda}:\ \gamma\in (0,\infty) \}$. give precise behavior U_{\gamma,q,\lambda}$ at infinity, distinguishing between $1\max\{q_{N,\theta},1\}$, $q_{N,\theta}=(N+2\theta+2)/(N-2)$. addition, $\theta\leq -2$ settle structure $\Omega\setminus \{0\}$, subject to $u|_{\partial\Omega}=0$, $\Omega$ smooth bounded domain containing zero, complementing works C\^{\i}rstea (Mem. Amer. Math. Soc. 227, 2014) Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).