In this paper,we study the existence of localized nodal solutions for Schrodinger-Poisson systems with critical growth{−ε^(2)Δv+V(x)v+λψv=v^(5)+μ|v|^(q−2)v,in R^(3),−ε^(2)Δψ=v^(2),in R^(3);v(x)→0,ψ(x)→0as|x...In this paper,we study the existence of localized nodal solutions for Schrodinger-Poisson systems with critical growth{−ε^(2)Δv+V(x)v+λψv=v^(5)+μ|v|^(q−2)v,in R^(3),−ε^(2)Δψ=v^(2),in R^(3);v(x)→0,ψ(x)→0as|x|→∞.We establish,for smallε,the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method,and employ some new analytical skills to overcome the obstacles caused by the nonlocal term φu(x)=1/4π∫R^(3)u^(2)(y)/|x−y|dy.Our results improve and extend related ones in the literature.展开更多
We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical ...We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.展开更多
We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of...We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of nonnegative ground state solutions is established.Our method relies upon the variational method and some analysis tricks.展开更多
This paper is concerned with the following fractional Schrodinger-Poisson system:{-(Δ)^(s)u+u+φu=λf(u)in R^(3)-(Δ)^(α)φu=u^(2)in R^(3)where s∈(3/4,1),α∈(0,1),λis a positive parameter,(-△)^(s),(-△)^(α)are ...This paper is concerned with the following fractional Schrodinger-Poisson system:{-(Δ)^(s)u+u+φu=λf(u)in R^(3)-(Δ)^(α)φu=u^(2)in R^(3)where s∈(3/4,1),α∈(0,1),λis a positive parameter,(-△)^(s),(-△)^(α)are fractional Laplacian operators.Under certain assumptions on f,we obtain the existence of at least one nontrivial solution of the system by using the methods of perturbation and Moser iterative method.展开更多
In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlin...In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.展开更多
Based on Nehari manifold, Schwarz symmetric methods and critical point theory, we prove the existence of positive radial ground states for a class of Schrodinger-Poisson systems in , which doesn’t require any symmetr...Based on Nehari manifold, Schwarz symmetric methods and critical point theory, we prove the existence of positive radial ground states for a class of Schrodinger-Poisson systems in , which doesn’t require any symmetry assumptions on all potentials. In particular, the positive potential is interesting in physical applications.展开更多
In this paper,a class of nonhomogeneous Schrodinger-Poisson systems with strong singularity are considered.Combining with the variational method and Nehari method,we obtain a positive solution for this system which im...In this paper,a class of nonhomogeneous Schrodinger-Poisson systems with strong singularity are considered.Combining with the variational method and Nehari method,we obtain a positive solution for this system which improves the recent results in the literature.展开更多
In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f...In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f satisfies some suitable conditions.Based on the Mountain pass theorem,we prove the existence of positive ground state solutions.展开更多
We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potent...We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.展开更多
We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artific...We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.展开更多
文摘In this paper,we study the existence of localized nodal solutions for Schrodinger-Poisson systems with critical growth{−ε^(2)Δv+V(x)v+λψv=v^(5)+μ|v|^(q−2)v,in R^(3),−ε^(2)Δψ=v^(2),in R^(3);v(x)→0,ψ(x)→0as|x|→∞.We establish,for smallε,the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method,and employ some new analytical skills to overcome the obstacles caused by the nonlocal term φu(x)=1/4π∫R^(3)u^(2)(y)/|x−y|dy.Our results improve and extend related ones in the literature.
基金the Science and Technology Project of Education Department in Jiangxi Province(GJJ180357)the second author was supported by NSFC(11701178).
文摘We study the following nonlinear fractional Schrodinger-Poisson system with critical growth:{(-△)sμ+μ+φμ=f(μ)+|μ|2s-2μ,x∈R3.(-△)tφ=μ2x∈R3,(0.1)where 0<s,t<1,2s+2t>3 and 2s=6/3-2s is the critical Sobolev exponent in 1R3.Under some more general assumptions on f,we prove that(0.1)admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.
基金National Natural Science Foundation of China(11471267)the first author was supported by Graduate Student Scientific Research Innovation Projects of Chongqing(CYS17084).
文摘We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|^p-1u,where the value of |x|→∞ lim Q(x)may not exist and Q may change sign.This means that the problem may have no limit problem.The existence of nonnegative ground state solutions is established.Our method relies upon the variational method and some analysis tricks.
文摘This paper is concerned with the following fractional Schrodinger-Poisson system:{-(Δ)^(s)u+u+φu=λf(u)in R^(3)-(Δ)^(α)φu=u^(2)in R^(3)where s∈(3/4,1),α∈(0,1),λis a positive parameter,(-△)^(s),(-△)^(α)are fractional Laplacian operators.Under certain assumptions on f,we obtain the existence of at least one nontrivial solution of the system by using the methods of perturbation and Moser iterative method.
基金supported by the National NaturalScience Foundation of China(12071170,11961043,11931012,12271196)supported by the excellent doctoral dissertation cultivation grant(2022YBZZ034)from Central China Normal University。
文摘In this paper,we study the following Schrodinger-Poisson system with critical growth:■We establish the existence of a positive ground state solution and a least energy sign-changing solution,providing that the nonlinearity f is super-cubic,subcritical and that the potential V(x)has a potential well.
文摘Based on Nehari manifold, Schwarz symmetric methods and critical point theory, we prove the existence of positive radial ground states for a class of Schrodinger-Poisson systems in , which doesn’t require any symmetry assumptions on all potentials. In particular, the positive potential is interesting in physical applications.
基金Supported by the Scientific Research Fund of Sichuan Provincial Education Department(18ZA0471)Fundamental Research Funds of China West Normal University(18B015)Innovative Research Team of China West Normal University(CXTD2018-8).
文摘In this paper,a class of nonhomogeneous Schrodinger-Poisson systems with strong singularity are considered.Combining with the variational method and Nehari method,we obtain a positive solution for this system which improves the recent results in the literature.
基金Supported by the Fundamental Research Funds of China West Normal University(No.18B015)Natural Science Foundation of Sichuan(No.23NSFSC1720).
文摘In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f satisfies some suitable conditions.Based on the Mountain pass theorem,we prove the existence of positive ground state solutions.
基金supported by Ministry of Education of Singapore grant R-146-000-120-112the National Natural Science Foundation of China(Grant No.11131005)the Doctoral Programme Foundation of Institution of Higher Education of China(Grant No.20110002110064).
文摘We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.
基金Singapore A*STAR SERC PSF-Grant No.1321202067National Natural Science Foundation of China Grant NSFC41390452the Doctoral Programme Foundation of Institution of Higher Education of China as well as by the Austrian Science Foundation(FWF)under grant No.F41(project VICOM)and grant No.I830(project LODIQUAS)and grant No.W1245 and the Austrian Ministry of Science and Research via its grant for the WPI.
文摘We study the computation of ground states and time dependent solutions of the Schr¨odinger-Poisson system(SPS)on a bounded domain in 2D(i.e.in two space dimensions).On a disc-shaped domain,we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion inθ,and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within O(M NlogN)arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases.Combined with the Poisson solver,a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively.Numerical results are shown to confirm the accuracy and efficiency.Also we make it clear that backward Euler sine pseudospectral(BESP)method in[33]can not be applied to 2D SPS simulation.
