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

Abstract

For $N\ge 3$, by the seminal paper of Brezis and Véron (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 $q\ge N/(N-2)$; 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 Véron (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 $\lambda >{\lambda }^{\ast }$, where ${\lambda }^{\ast }=\Theta (N-2-\Theta )$ with $\Theta =(\theta +2)/(q-1)$. We show that (a) if $\lambda >(N-2{)}^2/4$, then $U_0(x)=(\lambda -{\lambda }^{\ast }{)}^{1/(q-1)}|x{|}^{-\Theta }$ is the only solution of (*) and (b) if ${\lambda }^{\ast }<\lambda \le (N-2{)}^2/4$, 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 $\theta \le -2$ 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\^ırstea (Mem. Amer. Math. Soc. 227, 2014) and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).

32 pages

References in corpus (2)

• On the best possible remaining term in the Hardy Inequality
• Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations