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

Florica C. Cîrstea and Maria Fărcăşeanu

Abstract

In this paper, for every $q>1$ and $\theta \in \mathbb{R}$, we prove that the nonlinear elliptic problem $$\begin{array}{}\text{(}\ast \text{)}& -\mathrm{\Delta }u-\lambda |x{|}^{-2}u+|x{|}^{\theta }{u}^q=0\text{ in }{\mathbb{R}}^N\setminus \{0\}\text{ with }u>0\end{array}$$ has a $C^1({\mathbb{R}}^N\setminus \{0\})$ solution if and only if $\lambda >{\lambda }^{\ast }$, where ${\lambda }^{\ast }=\mathrm{\Theta }(N-2-\mathrm{\Theta })$ with $\mathrm{\Theta }=(\theta +2)/(q-1)$. We show that (a) if $\lambda >(N-2{)}^2/4$, then $U_0(x)=(\lambda -{\lambda }^{\ast }{)}^{1/(q-1)}|x{|}^{-\mathrm{\Theta }}$ is the only solution of ($\ast$) and (b) if ${\lambda }^{\ast }<\lambda \le (N-2{)}^2/4$, then all solutions of ($\ast$) are radially symmetric and their total set is $U_0\cup \{U_{\gamma ,q,\lambda }:\gamma \in (0,\mathrm{\infty })\}$. 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 ($\ast$) in $\mathrm{\Omega }\setminus \{0\}$, subject to $u{|}_{\partial \mathrm{\Omega }}=0$, where $\mathrm{\Omega }$ is a smooth bounded domain containing zero, complementing the works of Cîrstea (Mem. Amer. Math. Soc. 227, 2014) and Wei–Du (J. Differential Equations 262(7):3864–3886, 2017).

Keywords: Isolated singularities, Hardy potential, nonlinear elliptic equations, sub-super-solutions.