Multiple solutions for Grushin operator without odd nonlinearity

TL;DR

This paper establishes the existence of multiple solutions for certain degenerate elliptic equations involving the Grushin operator, without requiring the nonlinearity to satisfy the Ambrosetti-Rabinowitz condition, using variational methods.

Contribution

It introduces new existence and multiplicity results for degenerate elliptic equations with sign-changing potentials and non-AR nonlinearities, extending previous literature.

Findings

Two solutions for nonhomogeneous problem via mountain pass and Ekeland's principles.

Infinitely many solutions for homogeneous problem when nonlinearity is odd.

Results improve upon existing literature in degenerate elliptic equations.

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.