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

manuscripta mathematica - Tập 172 Số 3 - Trang 933-970 - 2023
Adachi, Shinji1, Ikoma, Norihisa2, Watanabe, Tatsuya3
1Department of Mathematical and Systems Engineering, Faculty of Engineering, Shizuoka University, Hamamatsu, Japan
2Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan
3Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Kyoto-City, Japan

Tóm tắt

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 $$\uplambda \rightarrow \infty $$ .

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