In this paper, we discuss the following elliptic equation: \[ -peratorname{div}(ǎrphi(x,abla u))=ambda f(x,u)\quadext{in}\quad\mathbb R^N, \] where the function $\varphi\colon\mathbb R^N\times\mathbb R^N\to\mathbb R^N$ is off type $|v|^{p-2}v$ with a real constant $p>1$ and $f\colon\mathbb R^N\times\mathbb R^N\to\mathbb R$ satisfies a Carathéodory condition.