  We consider quasilinear elliptic problems of the form \[
-\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi
(|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$
satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$
exhibits a finite or infinite potential well in the sense that $V(x)$ tends to
its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are
obtained by variational methods. When $V_{\infty }=+\infty$, a compact
embedding from a suitable subspace of $W^{1,\Phi }(\mathbb{R}^{N})$ into
$L^{\Phi}(\mathbb{R}^{N})$ is established, which enables us to get infinitely
many solutions for the case that $f$ is odd. For the case that $V(x)=\lambda
a(x) + 1$ exhibits a steep potential well controlled by a positive parameter
$\lambda$, we get nontrivial solutions for large $\lambda$.
