Multiple sign-changing solutions for fractional Schrödinger equations involving critical or supercritical exponent

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Abstract

In this article, we focus on the following fractional Schrödinger equation involving critical or supercritical exponent (P)(Δ)su+λV(x)u=|u|p2u+Q(x)|u|q2u,xRN,where 0<s<1, (Δ)s denotes the fractional Laplacian of order s, λ1, N>2s, 2<p<2sq and 2s=2NN2s. Under suitable assumptions on V(x) and Q(x), we prove that the above equation possesses k pairs of sign-changing solutions for large λ by using of truncation technique.

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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 Jλ(u)=12RN|ξ|2s|uˆ(ξ)|2dξ+12λRNV(x)u2dx1pRN|u|pdx1qRNQ(x)|u|qdx.We cannot apply variational methods directly because the functional Jλ is not well defined on Hs(RN) unless q=2s. In order to overcome this difficulty, we use a truncation technique to study Eq. (P). Let φC0(RN) be a cut-off function such that φ(t)=1 if |t|1, φ(t)=0 if |t|2, φ is an even function and decreasing for t

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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 .

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