In this article, we study the existence of nontrivial weak solutions for the following boundary value problem \(\begin{aligned} \begin{array}{cl} -\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}u=f(x,y,z,u)& \qquad \text {in}\quad \Omega , \\ u=0 & \qquad \text {on}\quad \partial \Omega , \end{array} \end{aligned}\) where \(\Omega \) is a bounded domain with smooth boundary in \(\mathbb {R}^N \ (N \ge 3), f(x,y,z,\xi ) \) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition, \(\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}\) is the subelliptic operator of the type \(\begin{aligned} \Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}: =\Delta _x +\Delta _y +\left| x\right| ^{2\alpha }\left| y\right| ^{2\beta }\left( \left| x\right| ^{\alpha _1}+\left| y\right| ^{\beta _1}\right) ^2\Delta _z;\\ x\in \mathbb R^{N_1}; y\in \mathbb R^{N_2}, z\in \mathbb R^{N_3}; N =N_1+N_2+N_3, \alpha , \beta , \alpha _1, \beta _1\ge 0. \end{aligned}\)