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 $\rho ,\lambda ,\theta \in \mathbb{R}$ are arbitrary, $N\ge 2$, $q>1$ 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\^ırstea and Fărcăşeanu (2021), where $\rho =(2-N)/2$ and $\tau =1$.

90 pages