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

This paper treats the existence of positive solutions of $$-\Delta u + V(x) u = \uplambda f(u)$$ in $${\mathbb {R}}^N$$ . Here $$N \ge 1$$ , $$\uplambda > 0$$ 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 (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family $$(u_\uplambda )_{\uplambda \ge \uplambda _0}$$ of positive solutions as $$\uplambda \rightarrow \infty $$ .