Normalized solutions of higher-order Schrödinger equations

Aliang Xia; Jianfu Yang

doi:10.3934/dcds.2019018

Article Contents

2019, Volume 39, Issue 1: 447-462. Doi: 10.3934/dcds.2019018

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Normalized solutions of higher-order Schrödinger equations

Aliang Xia^, and
Jianfu Yang

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author: Aliang Xia
* Corresponding author: Aliang Xia

Received: February 28, 2018

Revised: June 30, 2018

Published: December 2018

The first author is supported by the Foundation of Jiangxi Provincial Education Department, No: GJJ160335, the NNSF of China, No: 11701239 and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. The second author is supported by the NNSF of China, Nos: 11671179 and 11771300

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Abstract

In this paper, we consider the existence of non-trivial solutions for the following equation

$\mathcal{H}_{0J}u = |u|^{p-2}u+λ u\;\;\;\;{\rm in}\,\,\mathbb{R}^3,\;\;\;\;\;\;\;\;(1)$

where $\mathcal{H}_{0J}$ is the higher-order Schrödinger operator with $J∈\mathbb{N}$, $2<p<\frac{4J+6}{3}$, and $λ∈\mathbb{R}$ is a parameter. Let $E(u)$ be the corresponding variational functional of problem (1). We look for solutions of equation (1) by finding minimizers of the minimization problem

$E_ρ = \inf\{E(u)|u∈ H^{J}(\mathbb{R}^3):\,\,\|u\|_{L^2(\mathbb{R}^3)} = ρ\}.$

We show that problem (1) admits at least a solution provided that in the case $J$ being odd, $2<p<3$ and $ρ>0$ small or $2+J<p<\frac{4J+6}{3}$ and $ρ>0$ large; and for the case $J$ being even, $3<p<\frac{4J+6}{3}$ and $ρ>0$ small.

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Mathematics Subject Classification: Primary: 35J30, 35J35.

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