Multiple solutions for Grushin operator without odd nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: (Pg) - λu + V (x)u = f(x,u) + g(x),in a N, and (P0) - lambda;u + V (x)u = K(x)f(x,u),in a N, where λ is the strongly degenerate operator, V (x) is allowed to be sign-changing, K C(a N, a), g: aN → a is a perturbation and the nonlinearity f(x,u) is a continuous function does not satisfy the Ambrosetti-Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland's variational principle, existence of two different solutions for (Pg) are obtained when f satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for (P0) if f is odd in u thanks an extension of Clark's theorem near the origin. So, our main results considerably improve results appearing in the literature.