@article{Luo2022, 
author = {Haijun Luo and Zhitao Zhang},
title = {Existence and stability of normalized solutions to the mixed dispersion nonlinear Schrödinger equations},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {8},
pages = {2871-2898},
keywords = {normalized solution, orbital stability, general nonlinearity, biharmonic nonlinear Schrödinger equations},
url = {https://www.sciopen.com/article/10.3934/era.2022146},
doi = {10.3934/era.2022146},
abstract = {We study the existence and orbital stability of normalized solutions of the biharmonic equation with the mixed dispersion and a general nonlinear term   γΔ2u−βΔu+λu=f(u),x∈RNwith a priori prescribed  L2-norm constraint  Sa:={u∈H2(RN):∫RN|u|2dx=a}, where  a>0,  γ>0,β∈R and the nonlinear term  f satisfies the suitable  L2-subcritical assumptions. When  β≥0, we prove that there exists a threshold value  a0≥0 such that the equation above has a ground state solution which is orbitally stable if  a>a0 and has no ground state solution if  a<a0. However, for  β<0, this case is more involved. Under an additional assumption on  f, we get the similar results on the existence and orbital stability of ground state. Finally, we consider a specific nonlinearity  f(u)=|u|p−2u+μ|u|q−2u,2<q<p<2+8/N,μ<0 under the case  β<0, which does not satisfy the additional assumption. And we use the example to show that the energy in the case  β<0 exhibits a more complicated nature than that of the case  β≥0.}
}