Multiple solutions for Grushin operator without odd nonlinearity

Abstract

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] 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 [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.