Year: 2026
Type: preprint
Abstract: We study positive weak super-solutions of the Lane-Emden inequality $\mathcal{L}u := \sum_{i=1}^{n}(-Δ_{p_i})^{s_i}u \ge u^q$ in $\mathbb{R}^N$, where each $(-Δ_{p_i})^{s_i}$ is a fractional $p_i$-Laplacian of order $s_i\in(0,1]$ and exponent $p_i\in(1,\infty)$, and $\max_i s_ip_i\mathfrak{q}_{\mathcal{L}}$ we construct explicit positive weak super-solutions of the form $A(λ+|x|)^{-κ}$. In other words, the Serrin exponent of the sum is the smallest of the Serrin exponents $N/κ_i$ of the single operators, that is, the one of the operator whose fundamental solution decays fastest. No structural assumption relating the different operators is needed. The endpoint $q=\mathfrak{q}_{\mathcal{L}}$ is left open. The local order $s_i=1$ is treated together with the nonlocal orders, and all the arguments are carried out in the weak setting.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223356938