In this paper we deal with quasilinear elliptic equations of the form \(\begin{aligned} -\operatorname {div}\left( |\nabla u|^{p-2}\nabla u+a(\varepsilon x)|\nabla u|^{q-2}\nabla u \right) +|u|^{p-2} u+a(\varepsilon x)|u|^{q-2}u&=f(u) \end{aligned}\) in \(\mathbb {R}^N\) , where \(0 \le a(\cdot )\in C\left( \mathbb {R}^N\right) \cap L^{\infty }\left( \mathbb {R}^N\right) \) , \(1<p<N\) , \(p<q<p^*=\frac{Np}{N-p}\) , \(\varepsilon >0\) is a parameter, and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a continuous function that grows superlinearly and subcritically which does not need to fulfill the Ambrosetti-Rabinowitz condition. Based on the Lusternik-Schnirelmann category we prove several existence results of constant-sign and sign-changing solutions to the problem above provided the parameter \(\varepsilon >0\) is sufficiently small.
