Multiple sign-changing solutions for fractional Schrödinger equations involving critical or supercritical exponent☆
Section snippets
Introduction and main results
The fractional Schrödinger equation is a fundamental equation in fractional quantum mechanics. It was introduced by Laskin(see [1]) as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths, where the Feynman path integral leads to the classical Schrödinger equation and the path integral over Lévy trajectories leads to the fractional Schrödinger equation. Differently from the case of the classical Laplacian operator, the usual analysis
Proof of Theorem 1.1
It is well known to us that a weak solution of problem (P) is a critical point of the following functional We cannot apply variational methods directly because the functional is not well defined on unless . In order to overcome this difficulty, we use a truncation technique to study Eq. (P). Let be a cut-off function such that if , if , is an even function and decreasing for
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This work is supported in part by the National Natural Science Foundation of China (12026227; 12026228; 11801153; 11801545) and the Yunnan Province Applied Basic Research for Youths (2018FD085) and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013) and the Yunnan Province Applied Basic Research for General Project (2019FB001) and Technology Innovation Team of University in Yunnan Province .