Modified Q-Laplacian Problem with Parameter and Exponential Nonlinearity on the Heisenberg Group

2026/07/16 by Ankit Mishra, Sarika Goyal, Divya Goel
#math.AP

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Abstract

In this article, we investigate the following modified quasilinear equation driven by the Q-Laplacian: \begincases -Δ_Q u - Δ_Q(|u|^2α) |u|^2α-2 u = λf(ξ,u) in Ω,
u = 0 on ∂Ω, \endcases where Δ_Q(⋅) := div_ℍ(|∇_ℍ(⋅)|^Q-2∇_ℍ(⋅)) denotes the Q-Laplacian on the Heisenberg group ℍ^N, Ω⊂ ℍ^N is a smooth bounded domain with boundary ∂Ω, f behaves like exponential growth in the sense of Moser-Trudinger, and α> (1)/(2). The objectives of the paper are twofold: first, to establish the existence of a nontrivial positive weak solution, and subsequently to obtain least energy nodal (sign-changing) solutions under both subcritical and critical exponential growth assumptions on the nonlinearity f. The analysis relies on a suitable change of variables that reduces the original quasilinear structure to a semilinear variational framework, together with critical point theory on appropriately defined Nehari-type manifolds. The results derived here appear to be genuinely new even in the classical Euclidean setting, thereby extending the existing theory for quasilinear Schrödinger-type equations to the sub-Riemannian context of the Heisenberg group.

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