Existence of nontrivial positive solutions for generalized quasilinear elliptic equations with critical exponent

In this paper, we are concerned with the existence of nontrivial positive solutions for the following generalized quasilinear elliptic equations with critical growth

$$-\mathrm{d}\mathrm{i}\mathrm{v}(g^p(u)|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)+g^{p-1}(u)g^{\prime }(u)|\mathrm{\nabla }u{|}^p+V(x)|u{|}^{p-2}u=h(x,u),x\in {\mathbb{R}}^N,$$

where $N\ge 3$, $1<p<N$. Under some suitable conditions, we prove that the above equation has a nontrivial positive solution by variational methods. To some extent, our results improve and supplement some existing relevant results.