Nodal solutions for Q-Laplacian problem with exponential nonlinearities on the Heisenberg group

Abstract

This paper deals with the existence of nodal solutions (sign-changing) to the following elliptic equation involving Q-Laplacian:{−ΔQu=λf(ξ,u)inΩ,u=0on∂Ω, where ΔQ(⋅):=divHn(|∇Hn(⋅)|HnQ−2∇Hn(⋅)) is the Q-Laplacian operator, Ω is an open, smooth bounded domain in the Heisenberg group Hn, Q=2n+2 is the homogeneous dimension of Hn, λ is a positive parameter, and nonlinear term f(ξ,u) behaves like exp⁡(β|s|QQ−1) as s→∞. Under suitable conditions on f, together with the constraint variational method and the quantitative deformation lemma, we obtain the existence, energy estimates and the convergence property of the least energy sign-changing solutions for the above problem in subcritical and critical exponential growth cases.