Multiple solutions for Grushin operator without odd nonlinearity

Abstract

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: \begin{eqnarray*} (P_g)\quad - \Delta_{\lambda} u + V(x) u = f(x,u)+g(x),\;\mbox{ in } \R^N,\; \end{eqnarray*} and \begin{eqnarray*} (P_0)\quad - \Delta_{\lambda} u + V(x) u = K(x)f(x,u),\;\mbox{ in } \R^N,\; \end{eqnarray*} where $\Delta_{\lambda}$ is the strongly degenerate operator, $V(x)$ is allowed to be sign-changing, $K\in C(\R^N,\R)$, $g:\R^N\to\R$ 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 $(P_g)$ are obtained when $f$ satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for $(P_0)$ 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.