Elliptic problems involving a diffusion operator of vectorial p&q-Laplacian type
Abstract
We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator ℒAu=−∑i=1N∂i ( ai(|∂i u|pi)|∂i u|pi−2 ∂i u), where the functions ai satisfy structural growth and monotonicity conditions of (pi,qi)-type. This framework contains, among other examples, the anisotropic p⃗-Laplacian, the orthotropic p&q-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, r=p1<pN, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave--convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz--Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration--compactness.
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BibTeX
@misc{carvalho2026elliptic,
title = {{Elliptic problems involving a diffusion operator of vectorial $p\&q$-Laplacian type}},
author = {M. L. M. Carvalho and J. R. Santos Jr. and H. W. G. Silva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10838},
eprint = {2610.10838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10838},
}
RIS
TY - UNPB
TI - Elliptic problems involving a diffusion operator of vectorial p&q-Laplacian type
AU - M. L. M. Carvalho
AU - J. R. Santos Jr.
AU - H. W. G. Silva
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10838
UR - https://arxiv.org/abs/2610.10838
AB - We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator ℒ_Au=−∑_i=1^N∂_i ( a_i(|∂_i u|^p_i)|∂_i u|^p_i−2 ∂_i u), where the functions a_i satisfy structural growth and monotonicity conditions of (p_i,q_i)-type. This framework contains, among other examples, the anisotropic p⃗-Laplacian, the orthotropic p&q-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, r=p_1<p_N, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave--convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz--Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration--compactness.
ER -