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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 *** 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)>***,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/*** proofs are mainly based on the profile decomposition and a sharp Gagliardo-Nirenberg type *** 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

MULTIPLICITY OF normalized solutions FOR THE FRACTIONAL SCHR?DINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH

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Acta Mathematica Scientia 2024年第3期44卷 997-1019页

作者：孟禹希贺小明 School of Mathematics and Statistics Beijing Institute of TechnologyBeijing100081China College of Science Minzu University of ChinaBeijing100081China

In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 normalized solutions by using the genus theory and the truncation technique;in the L^(2)-supercritical case,we obtain a couple of normalized solutions by developing a fiber *** both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness *** results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.

关键词： fractional Schrodinger-Poisson system normalized solutions variational methods L^(2)-subcritical L^(2)-supercritical

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∈*** 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→*** also consider the case where I(x)may tend to-1 at infinity and the existence of multiple *** 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

The Complementarity of normalized solutions for Kirchhoff Equations with Mixed Nonlinearity

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Journal of Mathematical Research with Applications 2024年第4期44卷 483-494页

作者： Lin XU Qilin XIE School of Mathematics and Statistics Guangdong University of Technology

In this paper, we study the existence of solutions for Kirchhoff equation■ with mass constraint condition■ where a, b, c > 0, μ ∈ R, 2 0 when(p, q) belongs to a certain domain in R2. We prove the existence of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods. We partially extend the results which have been studied.

关键词： normalized solutions Kirchhoff type equation mixed nonlinearty

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

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Acta Mathematicae Applicatae Sinica 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 *** 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 *** 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