On the Lane–Emden–Matukuma equation −Δu=(1+|x|2)−σuα$-\Delta u = (1+|x|^2){}^{-\sigma } u^\alpha$ in Rn$\mathbf {R}^n$ with σ>1$\sigma >1$

Cao Thanh Tinh

Mathematische Nachrichten, 2025, vol. 298, issue 8, 2459-2475

Abstract: Of interest in this work is the following equation: −Δu=11+|x|2σuαinRn$$\begin{equation*} \hspace*{100pt}-\Delta u = {\left(\frac{1}{1+|x|^2}\right)}^\sigma u^\alpha \quad \text{in } \mathbf {R}^n \end{equation*}$$with n≥3$n \ge 3$, σ>1$\sigma > 1$, and α∈R$\alpha \in \mathbf {R}$. Our choice of this mathematical model, called Lane–Emden–Matukuma equation, is to provide a natural interpolation of the Lane–Emden equation corresponding to the case σ=0$\sigma =0$ and the Matukuma equation corresponding to the case σ=1$\sigma =1$. This is a continuation of our earlier work in which the case σ≤1$\sigma \le 1$ was studied. In this work, we are interested in non‐negative, non‐trivial, C2$C^2$‐solutions to the equation in the case σ>1$\sigma >1$ and α∈R$\alpha \in \mathbf {R}$. Our main result indicates that the equation always admits at least one solution unless n=3$n=3$, α=1$\alpha =1$, and 1

Date: 2025
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