  This paper treats the existence of positive solutions of $-\Delta u + V(x) u
= \lambda f(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $\lambda > 0$ is a
parameter and $f(u)$ satisfies conditions only in a neighborhood of $u=0$. We
shall show the existence of positive solutions with potential of trapping type
or $\mathcal{G}$-symmetric potential where $\mathcal{G} \subset O(N)$. Our
results extend previous results as well as we also study the asymptotic
behavior of a family $(u_\lambda)_{\lambda \geq \lambda_0}$ of positive
solutions as $\lambda \to \infty$.
