A new mathematical analysis reports a nontrivial solution for λ > 0 and no nontrivial solution for λ ≤ 0 within a specified transformed class, provided the combined assumptions hold.
The result appears in an arXiv version 1 preprint dated 26 Aug 2026. The work studies weak solutions written as u = g(v), with v in S^1_2(H^N), rather than an empirical participant sample.
A sign split at the heart of the problem
The equation is studied on the Heisenberg group H^N, and the target solutions are written as u = g(v), with v belonging to the space S^1_2(H^N). This transformed class defines the weak solutions addressed by the theorems.
The existence theorem applies at the critical exponent p = 2αQ* and requires the other exponent to satisfy max{4α, q*} < q < 2αQ*. It also requires the potential term to meet condition (V1) together with either (V2) or (V2′). Under those conditions, the paper asserts that at least one nontrivial solution exists.
The opposing theorem uses stated potential conditions and an odd nonlinearity whose primitive F obeys f(s)s ≥ 2αQ*F(s), along with conditions labelled (f1) and (f2). It also imposes (V3), including a lower bound on the derivative of the potential along an anisotropic dilation generator. Under those assumptions, no nontrivial weak solution exists in the transformed class.
How the proof gets past a difficult threshold
For the existence result, the authors use the dual change of variable u = g(v) to turn the quasilinear problem into a semilinear equation on S^1_2(H^N). They then use mountain-pass geometry, a variational setup for locating an energy critical point, and bounded Cerami sequences in the proof.
The proof combines a minimax estimate below the critical Sobolev threshold with truncated extremal functions and a Lions-type vanishing lemma. A translation argument is then used to obtain a nontrivial weak limit.
A key algebraic feature appears in the Pohozaev–Nehari pencil, an identity-based expression used in the analysis. At γ = 2αQ*, the quasilinear-energy term drops out exactly. The paper identifies this exponent as the point where that part of the energy disappears from the resulting balance.
What the result does—and does not—settle
Taken together, under the combined assumptions (V1), (V2) and (V3), with p = 2αQ* and max{4α, q*} < q < 2αQ*, the paper gives existence for λ > 0 and nonexistence for λ ≤ 0 within the transformed solution class. It also explicitly states that the purely critical case λ = 0 has no nontrivial solution.
Those conclusions are conditional, not general statements about every potential or every weak solution. The existence result is restricted to the stated critical exponent, q range and potential classes, while the nonexistence result is restricted to its own potential, nonlinearity, dilation and solution-class assumptions. The theorems concern solutions of the form u = g(v), with v in S^1_2(H^N).
The paper also reports that every weak solution is bounded and decays exponentially in the Koranyi gauge. As a mathematical analysis of a specified solution class, it offers conditional theorem conclusions rather than participant estimates.
The authors report receiving no funds, grants or other support for preparing the manuscript and report no conflict of interest. The supplied document is an arXiv version 1 preprint dated 26 Aug 2026.
Paper data and sources
Original title: Critical Quasilinear Schrödinger Equations on the Heisenberg Group: Existence and Nonexistence
Authors: Ankit Mishra, Divya Goel
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text