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THE EXISTENCE AND STABILITY OF normalized solutions FOR A BI-HARMONIC NONLINEAR SCHR?DINGER EQUATION WITH MIXED DISPERSION

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Acta Mathematica Scientia 2023年第2期43卷 539-563页

作者：罗庭健郑世骏朱世辉 School of Mathematics and Information Sciences Guangzhou UniversityGuangzhou 510006China Department of Mathematical Sciences Georgia Southern UniversityStatesboro 30460-8093USA School of Mathematical Sciences Sichuan Normal UniversityChengdu 610066China

In this paper,we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrodinger equation with aμ-Laplacian term(BNLS).Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term.Denoting by Qpthe ground state for the BNLS withμ=0,we prove that in the mass-subcritical regime p∈(1,1+8/d),there exist orbit ally stable ground state solutions for the BNLS when p∈(-λ0,∞)for someλ0=λ0(p,d,‖Qp‖L2)>0.Moreover,in the mass-critical case p=1+8/d,we prove the orbital stability on a certain mass level below‖Q*‖L2,provided thatμ∈(-λ1,0),where■and Q*=Q1+8/d.The proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type inequality.Our treatment allows us to fill the gap concerning the existence of the ground states for the BNLS when p is negative and p∈(1,1+8/d].

关键词： elliptic equations bi-harmonic operator normalized solutions profile decomposition

Localization of normalized solutions for saturable nonlinear Schr?dinger equations

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Science China Mathematics 2023年第11期66卷 2495-2522页

作者： Xiaoming Wang Zhi-Qiang Wang Xu Zhang School of Mathematics&Computer Science Shangrao Normal UniversityShangrao 334001China College of Mathematics and Statistics Fujian Normal UniversityFuzhou 350117China Department of Mathematics and Statistics Utah State UniversityLoganUT 84322USA School of Mathematics and Statistics Central South UniversityChangsha 410083China

In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.

关键词： saturable nonlinear Schrodinger equation normalized solutions semiclassical states local maximum potential

Non-existence of Multi-peak solutions to the Schrödinger-Newton System with L2-constraint

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应用数学学报:英文版 2023年第4期39卷 868-877页

作者： Qing GUO Li-xiu DUAN College of Science Minzu University of ChinaBeijing 100081China

In this paper, we are concerned with the the Schrödinger-Newton system with L^(2)-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at k different critical points of V(x) under certain assumptions on asymptotic behavior of V(x) and its first derivatives near these points. Especially, the critical points of V(x) in this paper must be degenerate.The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term.

关键词： Schrödinger-Newton equation Non-existence Multi-peak solutions normalized solutions Pohozaev identity

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带位势的基尔霍夫方程规范解的存在性

Chinese Quarterly Journal of Mathematics 2023年第2期38卷 196-209页

作者：梁艳霞 School of Mathematics and Statistics Guangdong University of TechnologyGuangzhou 510520China

In this paper we discuss the following Kirchhoff equation(−a+b R _(R3)|∇u|^(2) dx∆u+V(x)u+λu=µ|u|^(q−2)u+|u|^(p−2)u in R^(3),R R^(3) u^(2) dx=c^(2),where a,b,µ and c are positive numbers,λis unknown and appears as a Lagrange multiplier,143normalized solution.Here we study the existence of normalized solution to mass supercritical Kirchhoff equation with potential via the minimax principle and Nehari-Pohozaev manifold.

关键词： Kirchhoff equation normalized solutions Variational method