Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity
Abstract
We investigate a class of mixed-order quasilinear elliptic equations in ℝN driven by the q-biharmonic and p-Laplacian operators, Δq2u−Δpu = λ(|u|rσ−2u)/|x|σ +|u|p*−2u +|u|q**−2u, where 1<q<N/2, 1<p<q*, p≠ q, q≠ p*, and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation Δq2u−Δpu=f(x,u), where f is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both u and the nonlinear flux |Δu|q−2Δu. This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.
Related
- Global c1,α regularity and fine behavior at critical points for quasilinear elliptic equations with Orlicz growth and unbounded drift
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- Local regularity for solutions to singular and degenerate quasilinear parabolic equations
BibTeX
@misc{gloss2026doubly,
title = {{Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity}},
author = {Elisandra Gloss and Artur Jorge Marinho and Kanishka Perera and Bruno Ribeiro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28260},
eprint = {2609.28260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28260},
}
RIS
TY - UNPB
TI - Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity
AU - Elisandra Gloss
AU - Artur Jorge Marinho
AU - Kanishka Perera
AU - Bruno Ribeiro
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28260
UR - https://arxiv.org/abs/2609.28260
AB - We investigate a class of mixed-order quasilinear elliptic equations in ℝ^N driven by the q-biharmonic and p-Laplacian operators, Δ_q^2u−Δ_pu = λ(|u|^r_σ−2u)/|x|^σ +|u|^p^*−2u +|u|^q^**−2u, where 1<q<N/2, 1<p<q^*, p≠ q, q≠ p^*, and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation Δ_q^2u−Δ_pu=f(x,u), where f is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both u and the nonlinear flux |Δu|^q−2Δu. This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.
ER -