Classification and symmetry of global solutions for nonlinear elliptic equations with mixed reaction terms

In this paper, we describe the set of all positive distributional $C^1({\mathbb{R}}^N\setminus \{0\})$-solutions of elliptic equations with mixed reaction terms of the form $${\mathbb{L}}_{\rho ,\lambda ,\tau }[u]:=\mathrm{\Delta }u-(N-2+2\rho )\frac{x\cdot \mathrm{\nabla }u}{\mathrm{\mid }x{\mathrm{\mid }}^2}+\lambda \frac{u^{\tau }\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{1-\tau }}{\mathrm{\mid }x{\mathrm{\mid }}^{1+\tau }}=\mathrm{\mid }x{\mathrm{\mid }}^{\theta }{u}^{q}in{\mathbb{R}}^N\setminus \{0\},$$ where $\rho ,\lambda ,\theta \in \mathbb{R}$ are arbitrary, $N\ge 2$, $q>1$ and $\tau \in [0,1)$. Defining $\beta =(\theta +2)\mathrm{/}(q-1)$ and $f_{\rho ,\lambda ,\tau }(t)=t(t+2\rho )+\lambda \mathrm{\mid }t{\mathrm{\mid }}^{1-\tau }$ for $t\in \mathbb{R}$, we show that the equation has positive solutions if and only if $f_{\rho ,\lambda ,\tau }(\beta )>0$. Under this condition, we provide existence and the exact asymptotic behaviour near zero and at infinity for all positive solutions. We obtain that all such solutions are radially symmetric. When $\theta <-2$ and $\rho ,\lambda \in \mathbb{R}$, we also find the precise local behaviour near zero for all positive solutions of our equation in $\mathrm{\Omega }\setminus \{0\}$, where $\mathrm{\Omega }$ is an open set containing $0$. By introducing the second term in ${\mathbb{L}}_{\rho ,\lambda ,\tau }[\cdot ]$ with $\rho \in \mathbb{R}$, we reduce the study to $\theta <-2$ via a modified Kelvin transform. We reveal new and surprising phenomena compared with the work of Cîrstea and Fărcăşeanu (2021), where $\rho =(2-N)\mathrm{/}2$ and $\tau =1$.