在该文中,作者研究下述二次非线性薛定谔方程组{−ϵ^(2)Δu_(1)+V_(1)(x)u_(1)=αu_(1)u_(2),x∈R^(N),−ϵ^(2)Δu_(2)+V_(2)(x)u_(2)=α/2u_(1)^(2)+βu_(2)^(2),x∈R^(N),其中2≤N<6,ϵ>0为小参数,α>0和α>β,位势Vi是正的,...在该文中,作者研究下述二次非线性薛定谔方程组{−ϵ^(2)Δu_(1)+V_(1)(x)u_(1)=αu_(1)u_(2),x∈R^(N),−ϵ^(2)Δu_(2)+V_(2)(x)u_(2)=α/2u_(1)^(2)+βu_(2)^(2),x∈R^(N),其中2≤N<6,ϵ>0为小参数,α>0和α>β,位势Vi是正的,V_(i),|∇V_(i)|∈L^(∞)(R^(N)).当ϵ趋于0时,应用有限维约化方法我们构造了该方程组集中在由位势函数Vi(x)(i=1,2)构成的一个新函数的非退化临界点的同步解.此外,应用反证法结合局部的Pohozaev恒等式和爆破分析技巧,还证明了单峰解的唯一性.该文的结果将[Gross M.Ann Inst H Poincar'e C Anal Non Lin'eaire,2002]中关于单个非线性薛定谔方程的单峰解的结果推广到了该模型.展开更多
In this paper,we study the existence of least energy solutions for the following nonlinear fractional Schrodinger–Poisson system{(−∆)^(s)u+V(x)u+φu=f(u)in R^(3),(−∆)^(t)φ=u^(2)in R^(3),where s∈(3/4,1),t∈(0,1).Und...In this paper,we study the existence of least energy solutions for the following nonlinear fractional Schrodinger–Poisson system{(−∆)^(s)u+V(x)u+φu=f(u)in R^(3),(−∆)^(t)φ=u^(2)in R^(3),where s∈(3/4,1),t∈(0,1).Under some assumptions on V(x)and f,using Nehari–Pohozaev identity and the arguments of Brezis–Nirenberg,the monotonic trick and global compactness lemma,we prove the existence of a nontrivial least energy solution.展开更多
We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential.By exploring some delicate energy estimates and studying decay properties of solution sequen...We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential.By exploring some delicate energy estimates and studying decay properties of solution sequences,we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when a↗a^(*):=‖Q‖_(2)^(2s),where Q is the unique positive radial solution of (-△)^(s)u+su-|u|2su=0 in R^(2).Based on the discussion of the concentration phenomenon,we prove the local uniqueness of minimizers by establishing a local Poho zaev identity and studying the blow-up estimates to the nonlocal operator(-△)^(s).展开更多
In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-s...In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-shaped and bounded domain Ω for s E （0, 1）. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritieal cases.展开更多
This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function w...This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−2.Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r0>0 and K˜(r0,y˜0)>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).展开更多
In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of K...In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of Kahler-Einstein metric. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don＇t know if a priori radially symmetric property holds in the complex case.展开更多
基金Supported by the National Natural Science Foundation of China(12261053)the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities Association(2019FH001-078,202101BA070001-132)Introduction of Talents Research Project of Kunming University(XJ20210020,YJL20019)。
文摘在该文中,作者研究下述二次非线性薛定谔方程组{−ϵ^(2)Δu_(1)+V_(1)(x)u_(1)=αu_(1)u_(2),x∈R^(N),−ϵ^(2)Δu_(2)+V_(2)(x)u_(2)=α/2u_(1)^(2)+βu_(2)^(2),x∈R^(N),其中2≤N<6,ϵ>0为小参数,α>0和α>β,位势Vi是正的,V_(i),|∇V_(i)|∈L^(∞)(R^(N)).当ϵ趋于0时,应用有限维约化方法我们构造了该方程组集中在由位势函数Vi(x)(i=1,2)构成的一个新函数的非退化临界点的同步解.此外,应用反证法结合局部的Pohozaev恒等式和爆破分析技巧,还证明了单峰解的唯一性.该文的结果将[Gross M.Ann Inst H Poincar'e C Anal Non Lin'eaire,2002]中关于单个非线性薛定谔方程的单峰解的结果推广到了该模型.
基金Supported by NSFC(No.12561023)partly by the Provincial Natural Science Foundation of Jiangxi,China(Nos.20232BAB201001,20202BAB211004)。
文摘In this paper,we study the existence of least energy solutions for the following nonlinear fractional Schrodinger–Poisson system{(−∆)^(s)u+V(x)u+φu=f(u)in R^(3),(−∆)^(t)φ=u^(2)in R^(3),where s∈(3/4,1),t∈(0,1).Under some assumptions on V(x)and f,using Nehari–Pohozaev identity and the arguments of Brezis–Nirenberg,the monotonic trick and global compactness lemma,we prove the existence of a nontrivial least energy solution.
基金supported by the Fundamental Research Program of Shanxi Province(202403021222126)supported by the Fundamental Research Program of Shanxi Province(202303021211056)supported by the National Natural Science Foundation of China(12071486)。
文摘We consider a constrained minimization problem arising in the fractional Schrödinger equation with a trapping potential.By exploring some delicate energy estimates and studying decay properties of solution sequences,we obtain the concentration behavior of each minimizer of the fractional Schrödinger energy functional when a↗a^(*):=‖Q‖_(2)^(2s),where Q is the unique positive radial solution of (-△)^(s)u+su-|u|2su=0 in R^(2).Based on the discussion of the concentration phenomenon,we prove the local uniqueness of minimizers by establishing a local Poho zaev identity and studying the blow-up estimates to the nonlocal operator(-△)^(s).
基金supported by NSFC(Grant No.11571176)the second author is supported by NSFC(Grant No.11571057)
文摘In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-shaped and bounded domain Ω for s E （0, 1）. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritieal cases.
基金Supported by NSFC(Grant Nos.12226324,11961043,11801226)。
文摘This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−2.Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r0>0 and K˜(r0,y˜0)>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).
文摘In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of Kahler-Einstein metric. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don＇t know if a priori radially symmetric property holds in the complex case.
