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.

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 Friday, September 4, 2020