Positive and nodal solutions for a parametric quasilinear Q-sub-Laplacian problem with critical exponential growth on the Heisenberg group
Abstract
In this article, we investigate the following modified quasilinear equation with parameter driven by the Q-subLaplacian: −ΔQ u − ΔQ(|u|2α) |u|2α−2 u = λf(ξ,u) & in Ω, [2mm] u = 0 & on ∂Ω, where ΔQ(·):= divℍ(|∇ℍ(·)|Q−2∇ℍ(·)) denotes the Q-subLaplacian on the Heisenberg group ℍN, Q=2N+2 is the homogeneous dimension, Ω⊂ ℍN is a smooth bounded domain, λ>0, α> 1/2, and f has critical or subcritical exponential growth of order exp(β|t|2αQ/(Q−1)). We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large λ, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables u=g(v) reduces the problem to a quasilinear problem whose energy functional is of class C1; the exponent $2αQ/(Q-1)$ arises from the growth of g. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.
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BibTeX
@misc{mishra2026positive,
title = {{Positive and nodal solutions for a parametric quasilinear $Q$-sub-Laplacian problem with critical exponential growth on the Heisenberg group}},
author = {Ankit Mishra and Sarika Goyal and Divya Goel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.14804},
eprint = {2607.14804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14804},
}
RIS
TY - UNPB
TI - Positive and nodal solutions for a parametric quasilinear Q-sub-Laplacian problem with critical exponential growth on the Heisenberg group
AU - Ankit Mishra
AU - Sarika Goyal
AU - Divya Goel
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2607.14804
UR - https://arxiv.org/abs/2607.14804
AB - In this article, we investigate the following modified quasilinear equation with parameter driven by the Q-subLaplacian: −Δ_Q u − Δ_Q(|u|^2α) |u|^2α−2 u = λf(ξ,u) & in Ω, [2mm] u = 0 & on ∂Ω, where Δ_Q(·):= div_ℍ(|∇_ℍ(·)|^Q−2∇_ℍ(·)) denotes the Q-subLaplacian on the Heisenberg group ℍ^N, Q=2N+2 is the homogeneous dimension, Ω⊂ ℍ^N is a smooth bounded domain, λ>0, α> 1/2, and f has critical or subcritical exponential growth of order exp(β|t|^2αQ/(Q−1)). We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large λ, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables u=g(v) reduces the problem to a quasilinear problem whose energy functional is of class C^1; the exponent $2αQ/(Q-1)$ arises from the growth of g. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.
ER -