This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ＋Rψ（x,ψ） in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solution...This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ＋Rψ（x,ψ） in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth.展开更多
In this paper, we investigate nonlinear Hamiltonian elliptic system {-△u＋b（x）· u＋（V（x）＋τ）u=K（x）g（v） in R^N,-△u-b（x）· v＋（V（x）＋τ）v=K（x）f（u） in R^N,u（x）→ and v（x）→0 as ｜x｜...In this paper, we investigate nonlinear Hamiltonian elliptic system {-△u＋b（x）· u＋（V（x）＋τ）u=K（x）g（v） in R^N,-△u-b（x）· v＋（V（x）＋τ）v=K（x）f（u） in R^N,u（x）→ and v（x）→0 as ｜x｜→∞2,where N ≥ 3, τ 〉 0 is a positive parameter and V, K are nonnegative continuous functions,f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing avariational setting, the existence of ground state solutions is obtained.展开更多
In this paper,the authors study ground states for a class of K-component coupled nonlinear Schrödinger equations with a sign-changing potential which is periodic or asymptotically periodic.The resulting problem e...In this paper,the authors study ground states for a class of K-component coupled nonlinear Schrödinger equations with a sign-changing potential which is periodic or asymptotically periodic.The resulting problem engages three major difficulties:One is that the associated functional is strongly indefinite,the second is that,due to the asymptotically periodic assumption,the associated functional loses the Z^(N)-translation invariance,many effective methods for periodic problems cannot be applied to asymptotically periodic ones.The third difficulty is singular potentialμ/(|x|)^(2),which does not belong to the Kato’s class.These enable them to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential.展开更多
This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity...This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].展开更多
We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R...We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R^3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e^(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|^(P-2)E with 2<p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.展开更多
基金supported by the Hunan Provincial Innovation Foundation for Postgraduate(CX2013A003)the NNSF(11171351,11361078)SRFDP(20120162110021)of China
文摘This article is concerned with the nonlinear Dirac equations-iδtψ=ich ∑k=1^3 αkδkψ-mc^2βψ＋Rψ（x,ψ） in R^3.Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth.
基金partially supported by the Honghe University Doctoral Research Program(XJ17B11)Yunnan Province Applied Basic Research for Youthsthe Yunnan Province Local University(Part)Basic Research Joint Project(2017FH001-013)
文摘In this paper, we investigate nonlinear Hamiltonian elliptic system {-△u＋b（x）· u＋（V（x）＋τ）u=K（x）g（v） in R^N,-△u-b（x）· v＋（V（x）＋τ）v=K（x）f（u） in R^N,u（x）→ and v（x）→0 as ｜x｜→∞2,where N ≥ 3, τ 〉 0 is a positive parameter and V, K are nonnegative continuous functions,f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing avariational setting, the existence of ground state solutions is obtained.
基金supported by the Natural Science Foundation of Hubei Province of China(No.2021CFB473)Hubei Educational Committee(No.D20161206)
文摘In this paper,the authors study ground states for a class of K-component coupled nonlinear Schrödinger equations with a sign-changing potential which is periodic or asymptotically periodic.The resulting problem engages three major difficulties:One is that the associated functional is strongly indefinite,the second is that,due to the asymptotically periodic assumption,the associated functional loses the Z^(N)-translation invariance,many effective methods for periodic problems cannot be applied to asymptotically periodic ones.The third difficulty is singular potentialμ/(|x|)^(2),which does not belong to the Kato’s class.These enable them to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential.
基金supported by National Natural Science Foundation of China(Grant No.11171351)
文摘This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].
基金supported by the National Science Centre of Poland (Grant No. 2013/09/B/ST1/01963)
文摘We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R^3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e^(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|^(P-2)E with 2<p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.
