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

Abstract

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 $$ \mathbb L_{\rho,\lambda,\tau}[u]:= \Delta u-(N-2+2\rho) \frac{x\cdot \nabla u}{|x|^2} +\lambda \frac{u^\tau |\nabla u|^{1-\tau}}{|x|^{1+\tau}}=|x|^\theta u^q\quad \mbox{in } \mathbb R^N\setminus \{0\}, $$ where $\rho,\lambda, \theta\in \mathbb R$ are arbitrary, $N\geq 2$, $q>1$ and $\tau\in [0,1)$. Defining $\beta=(\theta+2)/(q-1)$ and $f_{\rho,\lambda,\tau}(t)=t\left(t+2\rho\right) +\lambda |t|^{1-\tau}$ 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 $\Omega\setminus \{0\}$, where $\Omega$ 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 C\^{\i}rstea and F\u{a}rc\u{a}\c{s}eanu (2021), where $\rho=(2-N)/2$ and $\tau=1$.