# American Institute of Mathematical Sciences

January  2019, 39(1): 447-462. doi: 10.3934/dcds.2019018

## Normalized solutions of higher-order Schrödinger equations

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

* Corresponding author: Aliang Xia

Received  March 2018 Revised  July 2018 Published  October 2018

Fund Project: 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

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 , 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
and
 $ρ>0$
small or
 $2+J and $ρ>0$large; and for the case $J$being even, $3
and
 $ρ>0$
small.
Citation: Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018
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