On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group

Let ℍ = ℝ × ℝ be the n−dimensional Heisenberg group, be its subelliptic gradient operator, and ρ (ξ) = ( |z| + t ) for ξ = (z, t) ∈ ℍ be the distance function in ℍ . Denote ℍ = ℍ , Q = 2n + 2 and Q′ = Q/(Q − 1). Let Ω be a bounded domain with smooth boundary in ℍ. Motivated by the Moser-Trudinger inequalities on the Heisenberg group, we study the existence of solution to a nonuniformly subelliptic equation of the form where f : Ω × ℝ → ℝ behaves like exp ( α |u| ) when |u| → ∞. In the case of Q−sub- Laplacian we will apply minimax methods to obtain multiplicity of weak solutions.