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

TL;DR

This paper classifies all positive solutions of a class of nonlinear elliptic equations with mixed reaction terms, establishing conditions for existence, symmetry, and asymptotic behavior, and revealing new phenomena compared to prior work.

Contribution

It provides a complete characterization of positive solutions, including existence criteria, symmetry, and asymptotic analysis, for a broad class of elliptic equations with mixed reaction terms.

Findings

Solutions exist if and only if a specific function exceeds zero.

All positive solutions are radially symmetric.

Detailed asymptotic behavior near zero and infinity is established.

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 $\tau=1$.