Quasilinear Schrödinger Critical Problem on the Heisenberg group \(ℍ^N\)

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

paper · pdf

Abstract

We study the existence of standing wave solutions for the following quasilinear Schrödinger equations with critical growth on the Heisenberg group -Δ_ℍ u +V(ξ)u-Δ_ℍ (|u|^2α)|u|^2α-2 u= λ|u|^q-2u + |u|^p-2u in ℍ^N where ℍ^N is Heisenberg group, Δ_ℍ is Kohn Laplacian operator, 4α<q<p ≤ 2αQ^*, \(Q^*= (2Q)/(Q-2)\) is the critical Folland--Stein exponent, λ and α are positive parameters, α> (1)/(2). By a suitable nonlinear change of variables, the quasilinear equation is transformed into a semilinear one, allowing the use of variational methods in the Folland--Stein Sobolev space S^1,2(ℍ^N). Applying the mountain pass theorem together with a concentration--compactness argument adapted to the sub-Riemannian framework, we establish the existence of a nontrivial solution.

Citations

• Existence of solitary waves in higher dimensions

Related

• Modified Q-Laplacian Problem with Parameter and Exponential Nonlinearity on the Heisenberg Group
• Standing wave solutions of a quasilinear Schrödinger equation in the small frequency limit
• Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in R^N
• The Existence of Global Solution for a Class of Semilinear Equations on Heisenberg Group
• Standing waves for quasilinear Schröinger equations with indefinite potentials