Abstract
We study the nonlinear problem \(-\Delta u + V(x)u = f(u), x \in \mathbb {R}^{N}, \lim _{ |x| \rightarrow \infty } u(x) = 0\), where the Schrödinger operator \(-\Delta + V\) is positive and f is asymptotically linear. Moreover, \(\lim _{|x| \rightarrow \infty } V(x) = \sigma _{0}\). We allow the interference of essential spectrum, i.e. \(\sup _{t \ne 0}f(t)/t \ge \sigma _{0}\). If \(\sup _{t \ne 0}2F(t)/t^{2} < \sigma _{0}\), the existence of four solutions will be proved by Morse theory. If \(\sup _{t \ne 0}2F(t)/t^{2} \ge \sigma _{0}\), we can find a positive solution when \(mes(\{x \in \mathbb {R}^{N}: V(x)> \sigma _{0}\}) > 0\).
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I would like to thank C. Li and S.J. Li for fruitful discussions and constant support during the preparation of this work.
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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.
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Song, L. Existence and multiplicity of solutions for Schrödinger equations with asymptotically linear nonlinearities allowing interaction with essential spectrum. Partial Differ. Equ. Appl. 3, 25 (2022). https://doi.org/10.1007/s42985-022-00162-7
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DOI: https://doi.org/10.1007/s42985-022-00162-7