Sharp existence and classification results for nonlinear elliptic equations in $\mathbb R^N\setminus\{0\}$ with Hardy potential (2009.00157v1)

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).