Li, Gui-Dong; Li, Yong-Yong; Tang, Chun-Lei A positive solution of asymptotically periodic Schrödinger equations with local superlinear nonlinearities. (English) Zbl 1463.35245 Electron. J. Qual. Theory Differ. Equ. 2020, Paper No. 30, 15 p. (2020). Summary: In this paper, we investigate the following Schrödinger equation \[-\Delta u+V(x)u=\lambda f(u) \quad {\text{ in}} \ \mathbb{R}^N,\] where \(N\geq 3\), \(\lambda>0\), \(V\) is an asymptotically periodic potential and the nonlinearity term \(f(u)\) is only locally defined for \(\vert u\vert \) small and satisfies some mild conditions. By using Nehari manifold and Moser iteration, we obtain the existence of positive solutions for the equation with sufficiently large \(\lambda\). MSC: 35J50 Variational methods for elliptic systems 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B09 Positive solutions to PDEs 35D30 Weak solutions to PDEs Keywords:Schrödinger equation; positive solution; locally defined nonlinearity; asymptotically periodic potential PDFBibTeX XMLCite \textit{G.-D. Li} et al., Electron. J. Qual. Theory Differ. Equ. 2020, Paper No. 30, 15 p. (2020; Zbl 1463.35245) Full Text: DOI