Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation

Abstract

This paper treats the existence of positive solutions of $-\Delta u+V(x)u=\lambda f(u)$ in ${\mathbb{R}}^N$. Here $N\ge 1$, $\lambda >0$ 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 $(u_{\lambda }{)}_{\lambda \ge {\lambda }_0}$ of positive solutions as $\lambda \to \infty$.

29 pages