Classification and symmetry of global solutions for nonlinear elliptic equations with mixed reaction terms
Huyuan Chen, Florica C. Cîrstea, Aleksandar Miladinovic
TL;DR
This work classifies all positive distributional $C^1(\mathbb{R}^N\setminus\{0\})$-solutions to a nonlinear elliptic equation with mixed reaction terms, $\mathbb L_{\rho,\lambda,\tau}[u]=|x|^{\theta}u^q$, by introducing the invariant scaling $\beta=(\theta+2)/(q-1)$ and the auxiliary function $f_{\rho}(t)=t(t+2\rho)+\lambda|t|^{1-\tau}$. A sharp existence criterion $f_{\rho}(\beta)>0$ is established, under which every positive solution is radially symmetric and exhibits precise asymptotics near $0$ and at infinity, with the solution families $U_{\rho,\beta}$, $u_{\infty,c}$, $u_b$, and $u_{b,\infty,c}$ capturing all viable behaviours. The analysis introduces a robust barrier/sub-solution framework, a novel iterative scheme for refining local profiles, and a modified Kelvin transform to relate near-zero and near-infinity regimes, revealing new phenomena for $\tau\in(0,1)$ beyond the classical $\tau=1$ case. The paper also constructs detailed asymptotic models via the homogeneous operator $\mathbb L_{\rho,\lambda,\tau}[\cdot]=0$ and derives log-corrected sharp bounds, culminating in a complete trichotomy for the near-zero behaviour in the critical parameter ranges. Overall, the results provide a comprehensive global classification, symmetry, and sharp asymptotics for a broad family of nonlinear elliptic problems with gradient-dependent terms, with potential extensions to quasilinear settings and related gradient-driven dynamics.
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 $τ=1$.