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 $$ \mathbb L_{ρ,λ,τ}[u]:= Δu-(N-2+2ρ) \frac{x\cdot \nabla u}{|x|^2} +λ\frac{u^τ|\nabla u|^{{1-τ}}{|x|^{{1+τ}}=|x|^{θu^q\quad}}} \mbox{in } \mathbb R^N\setminus {0}, $$ where $ρ,λ, θ\in \mathbb R$ are arbitrary, $N\geq 2$, $q&gt;1$ and $τ\in [0,1)$. Defining $β=(θ+2)/(q-1)$ and $f_{ρ,λ,τ}(t)=t\left(t+2ρ\right) +λ|t|<sup>{1-τ}$ for $t\in \mathbb R$, we show that the equation has positive solutions if and only if $f_{ρ,λ,τ}(β)&gt;0$. 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 $θ&lt;-2$ and $ρ,λ\in \mathbb R$, we also find the precise local behaviour near zero for all positive solutions of our equation in $Ω\setminus {0}$, where $Ω$ is an open set containing $0$. By introducing the second term in $\mathbb L_{ρ,λ,τ}[\cdot]$ with $ρ\in \mathbb R$, we reduce the study to $θ&lt;-2$ via a modified Kelvin transform. We reveal new and surprising phenomena compared with the work of Cîrstea and Fărcăşeanu (2021), where $ρ=(2-N)/2$ and $τ=1$.