Classification and symmetry of global solutions for nonlinear elliptic equations with mixed reaction terms
Authors
Huyuan Chen
Florica C. Cîrstea
Aleksandar Miladinovic
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, $N\geq 2$, $q>1$ and $τ\in [0,1)$. Defining $β=(θ+2)/(q-1)$ and $f_{ρ,λ,τ}(t)=t\left(t+2ρ\right) +λ|t|^{1-τ}$ for $t\in \mathbb R$, we show that the equation has positive solutions if and only if $f_{ρ,λ,τ}(β)>0$. 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 $θ<-2$ 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 $θ<-2$ 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 $τ=1$.
