Asymptotic Behavior of Radial Solutions to Singular p-Laplacian Equations as p→1
Abstract
We study the singular limit p ↓ 1 for positive radial entire solutions up of −Δp up=Fp(|x|, up,|∇ up|) up−βp in ℝn, where n ≥ 2,1<p<min {2, √n}, and 0 ≤ βp ≤ p−1. Assuming that Fp → F1 locally uniformly , βp/(p−1) → c ∈[0,1] as p ↓ 1, we prove that, after passing to a subsequence, upj → u in Cloc (ℝn) and ∇ upj stackrel*rightharpoonup ∇ u in Lloc ∞(ℝn ; ℝn), where u ∈ W1, ∞(ℝn) is radial, and solves −Δ1 u=K(|x|) in ℝn . On {x ∈ ℝn | G(|x|)<1}, we prove that K(|x|)=w−c F1(|x|, u(x), 0) and that ∇ u=0 a.e., where upipi−1 → w. On the other hand, we show that if ρ↦ F1(r, t, ρ) is affine for every r and t, then K(|x|)= w−c F1(|x|, u(x),|∇ u(x)|) a.e. in ℝn. For n=1 and 1<p<2, we study d/( d x)(|up′|p−2 up′)=Fp(|x|, up,|up′|) up−βp in ℝ and obtain analogous results as p ↓ 1.
Related
- Existence, multiplicity and asymptotic behavior of positive solutions to (p,q)-Laplacian elliptic systems
- Asymptotic analysis and uniqueness of blowup solutions of non-quantized singular mean field equations
- Existence and classification of solutions for singular 1-Laplacian equations with gradient terms
- Asymptotic behavior of bounded Mild Solutions to some fractional differential equations involving the ψ-Hilfer derivative
- Local behavior of solutions to inhomogeneous singular parabolic p-Laplace equations
BibTeX
@misc{apaza2026asymptotic,
title = {{Asymptotic Behavior of Radial Solutions to Singular $p$-Laplacian Equations as $p\to1$}},
author = {Juan Pablo Alcon Apaza},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17511},
eprint = {2609.17511},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17511},
}
RIS
TY - UNPB
TI - Asymptotic Behavior of Radial Solutions to Singular p-Laplacian Equations as p→1
AU - Juan Pablo Alcon Apaza
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17511
UR - https://arxiv.org/abs/2609.17511
AB - We study the singular limit p ↓ 1 for positive radial entire solutions u_p of −Δ_p u_p=F_p(|x|, u_p,|∇ u_p|) u_p^−β_p in ℝ^n, where n ≥ 2,1<p<min {2, √n}, and 0 ≤ β_p ≤ p−1. Assuming that F_p → F_1 locally uniformly , β_p/(p−1) → c ∈[0,1] as p ↓ 1, we prove that, after passing to a subsequence, u_p_j → u in C_loc (ℝ^n) and ∇ u_p_j stackrel*rightharpoonup ∇ u in L_loc ^∞(ℝ^n ; ℝ^n), where u ∈ W^1, ∞(ℝ^n) is radial, and solves −Δ_1 u=K(|x|) in ℝ^n . On {x ∈ ℝ^n | G(|x|)<1}, we prove that K(|x|)=w^−c F_1(|x|, u(x), 0) and that ∇ u=0 a.e., where u_p_i^p_i−1 → w. On the other hand, we show that if ρ↦ F_1(r, t, ρ) is affine for every r and t, then K(|x|)= w^−c F_1(|x|, u(x),|∇ u(x)|) a.e. in ℝ^n. For n=1 and 1<p<2, we study d/( d x)(|u_p^′|^p−2 u_p^′)=F_p(|x|, u_p,|u_p^′|) u_p^−β_p in ℝ and obtain analogous results as p ↓ 1.
ER -