Nontrivial Solutions to Boundary Value Problems for Semilinear Δ_γ-Differential Equations

In this article, we study the existence of nontrivial weak solutions for the following boundary value problem:

−Δ_γu = f(x, u) in Ω, u = 0 on ∂Ω,

where Ω is a bounded domain with smooth boundary in , Ω ∩ {x_j = 0} ≠ ∅ for some j, Δ_γ is a subelliptic linear operator of the type

where γ(x) = (γ₁(x), γ₂(x), …, γ_N(x)) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f(x, ξ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.