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 \[-\Delta u-\lambda \,|x|^{-2}\,u+|x|^{\theta}u^q=0 \quad \text{ in \(\mathbb R^N\setminus \{0\}\) with \(u>0\)} \tag{\(*\)}\] 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 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.