PreprintCC BY-NC-ND 4.0arXiv
Sobolev Inequalities and the Existence of Solutions to Degenerate p-Poisson Equations
Abstract
In this paper we study an equivalence between the existence of a Sobolev inequality without gain, |φ|Lp(v,Ω) ≤ S(p,1) | √Q∇ φ|Lp(Ω), that holds for smooth functions of compact support and the existence of a degenerate weak solution (u,∇ u)∈ QH1,p0(v,Ω) to a Dirichlet problem for the p-Laplacian with a zero order term: −v−1Div(|√Q∇ u|p−2Q∇ u)+F|u|p−2u = |f|p−2f − v−1Div(v|g|p−2gt), x ∈ Ω, and u =0, x ∈ ∂ Ω, More precisely, we use the Sobolev inequality to prove the existence of a degenerate weak solution to this equation and then use the existence of such a solution to produce a Sobolev inequality. Moreover, we show that solutions are unique.
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BibTeX
@misc{cruzuribe2026sobolev,
title = {{Sobolev Inequalities and the Existence of Solutions to Degenerate $p$-Poisson Equations}},
author = {David Cruz-Uribe and Feyza Dal and Scott Rodney},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22602},
eprint = {2609.22602},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22602},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sobolev Inequalities and the Existence of Solutions to Degenerate p-Poisson Equations
AU - David Cruz-Uribe
AU - Feyza Dal
AU - Scott Rodney
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22602
UR - https://arxiv.org/abs/2609.22602
AB - In this paper we study an equivalence between the existence of a Sobolev inequality without gain, |φ|_L^p(v,Ω) ≤ S(p,1) | √Q∇ φ|_L^p(Ω), that holds for smooth functions of compact support and the existence of a degenerate weak solution (u,∇ u)∈ QH^1,p_0(v,Ω) to a Dirichlet problem for the p-Laplacian with a zero order term: −v^−1Div(|√Q∇ u|^p−2Q∇ u)+F|u|^p−2u = |f|^p−2f − v^−1Div(v|g|^p−2gt), x ∈ Ω, and u =0, x ∈ ∂ Ω, More precisely, we use the Sobolev inequality to prove the existence of a degenerate weak solution to this equation and then use the existence of such a solution to produce a Sobolev inequality. Moreover, we show that solutions are unique.
ER -