Topological invariants for anisotropic quasilinear elliptic systems: A Poincaré-Hopf formula
Abstract
We consider the functional Iδ,Ψ1,Ψ2:X → ℝ defined for any z=(u,v) ∈ X as Iδ,Ψ1,Ψ2(z) & = ∫Ω Ψ1(∇ u) dx + ∫Ω Ψ2(∇ v ) dx & − ∫Ω H(δ,x,u(x),v(x)) dx, where Ω is a smooth bounded domain of ℝN, Ψ1, Ψ2: ℝN → ℝ are convex functions satisfying suitable conditions for 1 < p, q < N, and the nonlinearity H is allowed to exhibit both subcritical and critical growth and may depend on a parameter δ∈ I ⊆ ℝ. Here, the functional space setting is given by the product space X :=W01,p(Ω)× W01,q(Ω) equipped with the norm |z|= |u|1,p + |v|1,q for any z=(u,v)∈ X, where | · |1,s is the usual norm in W1,s0(Ω). In this paper we prove that Iδ,Ψ1,Ψ2' is of class (S)+ and we infer that each isolated critical point of Iδ,Ψ1,Ψ2 has critical groups of finite type and a Poincaré-Hopf formula holds.
Related
- Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
- Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms
- On a Nonuniformly Coercive Quasilinear p(·)-Elliptic Neumann Problem with L1 Data
- Local and global C1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type
- Liouville Theorems for Stable Solutions to Anisotropic Quasilinear Equations
BibTeX
@misc{borgia2026topological,
title = {{Topological invariants for anisotropic quasilinear elliptic systems: A Poincaré-Hopf formula}},
author = {Natalino Borgia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40145},
eprint = {2609.40145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40145},
}
RIS
TY - UNPB
TI - Topological invariants for anisotropic quasilinear elliptic systems: A Poincaré-Hopf formula
AU - Natalino Borgia
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40145
UR - https://arxiv.org/abs/2609.40145
AB - We consider the functional I_δ,Ψ_1,Ψ_2:X → ℝ defined for any z=(u,v) ∈ X as I_δ,Ψ_1,Ψ_2(z) & = ∫_Ω Ψ_1(∇ u) dx + ∫_Ω Ψ_2(∇ v ) dx & − ∫_Ω H(δ,x,u(x),v(x)) dx, where Ω is a smooth bounded domain of ℝ^N, Ψ_1, Ψ_2: ℝ^N → ℝ are convex functions satisfying suitable conditions for 1 < p, q < N, and the nonlinearity H is allowed to exhibit both subcritical and critical growth and may depend on a parameter δ∈ I ⊆ ℝ. Here, the functional space setting is given by the product space X :=W_0^1,p(Ω)× W_0^1,q(Ω) equipped with the norm |z|= |u|_1,p + |v|_1,q for any z=(u,v)∈ X, where | · |_1,s is the usual norm in W^1,s_0(Ω). In this paper we prove that I_δ,Ψ_1,Ψ_2' is of class (S)_+ and we infer that each isolated critical point of I_δ,Ψ_1,Ψ_2 has critical groups of finite type and a Poincaré-Hopf formula holds.
ER -