Multiple Solutions for the Klein-Gordon-Maxwell System with Steep Potential Well

In this paper,we concern the Klein-Gordon-Maxwell system with steep potential well{-△u+(λa(x)+1)u-(2w+φ)φu=f(x,u),in R^3-△φ=-(w+)u^2,in R^3 Without global and local compactness,we can tell the difference of multiple solutions from their norms in Lp(R3).     

出现在双色谱中的非线性双曲型守恒律组整体光滑解的存在性(英文)

本文讨论了出现在双色谱中的非线性双曲型守恒律组的如下Cauchy问题{ut+(u/1+u+v)x=0,vt+(v/1+u+v)x=0,初值为u(x,0)=u0(x),v(x,0)=v0(x)的整体光滑解的存在性和唯一性.分析过程基于对角化方法和特征线法.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

临界Hartree方程组基态解的存在性

本文考虑临界耦合的Hartree方程组{-△+λu=∫Ω|u(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,-△+νu=∫Ω|ν(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,其中Ω是RN中带有光滑边界的有界区域,N≥3,λ,v是常数,且满足λ,v>-λ1(Ω),λ1(Ω)是(-△,H01(Ω))的第一特征值,β> 0是耦合参数,临界指标2μ*=(2N-μ)/(N-2)来源于Hardy-LittlewoodSobolev不等式,利用变分的方法证明了临界Hartree方程组基态正解的存在性.     

带有Logistic源的吸引-排斥趋化性系统的整体有界性和渐近行为

该文研究了有界区域ΩR~N(N≥1)中,齐次Neumann边值条件下带有Logistic源的吸引-排斥趋化性系统u_t=Δu-▽·(u▽v)+μ_1u(1-u),0=Δv+w-v,w_t=Δw+▽·(w▽z)+μ_2w(1-w),0=△z-z+u,其中μ_1,μ_20.证明了对任何非负初值u_0(x),w_0(x)∈C(Ω),解(u(·,t),v(·,t),w(·,t),z(·,t))整体有界.此外,如果μ_1,μ_21/16,那么当t→∞时,解(u(·,t),u(·,t),w(·,t),z(·,t))在L~∞模意义下渐近收敛于常数平衡解(1,1,1,1).     

Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper,by using the idea of category,we investigate how the shape of the graph of h(x)affects the number of positive solutions to the following weighted nonlinear elliptic system:-div(|x|-2au)-μu|x|2(a+1)=αα+βh(x)|u|α-2|v|βu|x|b2*(a,b)+λK1(x)|u|q-2u,in,-div(|x|-2av)-μv|x|2(a+1)=βα+βh(x)|u|α|v|β-2v|x|b2*(a,b)+σK2(x)|v|q-2v,in,u=v=0,on,where 0∈is a smooth bounded domain in RN(N 3),λ,σ0 are parameters,0μμa(N-2-2a2)2;h(x),K1(x)and K2(x)are positive continuous functions in,1 q2,α,β1 andα+β=2*(a,b)(2*(a,b)2N N-2(1+a-b),is critical Sobolev-Hardy exponent).We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters(λ,σ)belongs to a certain subset of R2.     

带有Logistic源的吸引-排斥趋化性系统的整体有界性和渐近行为北大核心CSCD

该文研究了有界区域ΩR^N(N≥1)中,齐次Neumann边值条件下带有Logistic源的吸引-排斥趋化性系统u_t=Δu-▽·(u▽v)+μ_1u(1-u),0=Δv+w-v,w_t=Δw+▽·(w▽z)+μ_2w(1-w),0=△z-z+u,其中μ_1,μ_2>0.证明了对任何非负初值u_0(x),w_0(x)∈C(Ω),解(u(·,t),v(·,t),w(·,t),z(·,t))整体有界.此外,如果μ_1,μ_2>1/16,那么当t→∞时,解(u(·,t),u(·,t),w(·,t),z(·,t))在L~∞模意义下渐近收敛于常数平衡解(1,1,1,1).     

On the existence of solutions for an elliptic system of equations with arbitrary order growth

Let BR be the ball centered at the origin with radius R in RN ( N ≥2). In this paper we study the existence of solution for the following elliptic systemu -△u+λu=p/(p + q)κ(| x |)) u(p-1)vq1,x ∈BR1,-△u+λu=p/(p + q)κ(| x |)) upv(q-1)1,x ∈BR1,u > 01,v > 01,x ∈ BR1,(u)/(v)=01,(v)/(v)=01,x ∈BRwhereλ > 0 , μ > 0 p ≥ 2, q ≥ 2,ν is the unit outward normal at the boundary BR . Under certainassumptions on κ ( | x | ), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.     

POSITIVE SOLUTIONS FOR WEAKLY COUPLED NONLINEAR ELLIPTIC SYSTEMS

In this article,we consider the existence of positive solutions for weakly cou-pled nonlinear elliptic systems {-△u+u (1+a(x))|u| p-1 u+μ|u| α-2 u|v|β+λv in R~N,-△v+v=(1+bx))|v|p-1v+μ|u|α|v|β-2v+λu in R N.(0.1) To find nontrivial solutions,we first investigate autonomous systems.In this case,results of bifurcation from semi-trivial solutions are obtained by the implicit function theorem.Next,the existence of positive solutions of problem(0.1) is obtained by variational methods.     

一类非线性Schrdinger-Maxwell方程基态解的存在性

本文研究了如下Schrdinger-Maxwell方程基态解的存在性问题{-△u+V(x)u+K(x)φ(x)u=b(x)|u|p-1u+λg(x,u)in R~3,-△φ=K(x)u~2in R~3,其中λ0,V(x)∈C~1(R~3,R),且V(x)0.△在K,g,b满足一定的假设条件下,且0p1时,利用变分法和临界点理论,获得了基态解的存在性.该结论推广了文献[7]的结果.     

高阶半线性抛物型方程组的生命跨度

本文研究高阶半线性抛物型方程组{ut+(-△)mu=|u|p, (t,x)∈R1+×RN, ut+(-△)mν=|u|q, (t,x)∈R1+×RN,u(0,x)=u0(x),v(0,x)=uo(x),x∈RN,其中m,p,q＞1.利用试验函数方法,首先推导一些积分不等式,然后对方程组爆破解的生命跨度[0,T)给出估计.     

临界或超临界增长分数阶Schrodinger-Poisson方程正解的存在性

本文研究如下分数阶Schrodinger-Poisson方程{(-△)^su+Vx(u)+K(x)φu=f(u)+λ|u|^q-2ux∈R³,(-△)^tφ=K(x)u²,x∈R³其中S∈(3/4,1),t∈(0,1),f是在原点超线性无穷远次临界的连续非线性项,指数q≥2_s^*=6/3-2x.当λ>0充分小时,我们利用变分方法证明上述问题正解的存在性.本文的主要贡献是处理了超临界情形.     

Multiplicity of solutions for quasilinear elliptic systems with singularity

In this paper, we study the existence of multiple solutions for the following quasilinear elliptic system:p*(t)|u-2β- △pu1-μ|-2u up1= α1u + β1-2|xp||xt|vβ2||u|u, x∈,|q*β- △qv-μ2 |v|q-2v αv(s)-2|2x|q=|x|sv + β2|uβ1||v2 |-2v, x∈,u(x) = v(x) = 0, x∈ .Multiplicity of solutions for the quasilinear problem is obtained via variational method.     

一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.     

Bakry-Emery型Ricci曲率下两类抛物方程正解的梯度估计

本文考虑完备黎曼流形上,在Bakry-Emery型Ricci曲率有下界的条件下两类抛物方程?u/?t=△Vu+au log u 和(△v-?/?t)u(x,t)+p(x,t)uβ(x,t)+q(x,t)u(x,t)=0正解的梯度估计,这里α,β ∈(R),△V(·):=△+(V,▽(·)).由于引入了 △V,相应地,在...     

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

In this article,we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity-△u+(λa(x)+1)u=(1/|x|α*F(u))f(u) in R~N,where N≥3,0 αmin{N,4},λ is a positive parameter and the nonnegative potential function a(x) is continuous.Using variational methods,we prove that if the potential well int(a~(-1)(0)) consists of k disjoint components,then there exist at least 2~k-1 multi-bump solutions.The asymptotic behavior of these solutions is also analyzed as λ→+∞.     

二阶差分系统正解的存在性

本文考虑了二阶差分系统-△~2u(t-1)=λf(v(t)),t∈[1,T]_z,-△~2v(t-1)=λg(u(t)),t∈[1,T]_z,u(0)=u(T+1)=0,v(0)=v(T+1)=0正解的存在性,其中f,g∈C([0,∞),R),λ0是参数.在f和g满足适当的假设条件下运用Schauder不动点定理证明了当λ充分大时差分系统正解的存在性.     

一类Klein-Gordon-Maxwell方程无穷多解的存在性

主要研究下面含有参数且带有凹凸非线性项的Klein-Gordon-Maxwell方程无穷多解的存在性问题:{-△u+V(x)u-(2ω+φ)φu=λa(x)f(x,u)+μb(x)g(x,u),在R~3,△φ=(ω+φ)u~2,在R~3.(*)其中λ,μ是参数,ω是一个常数,且ω0.u,φ:R~3→R,V:R~3→R.在对V,a,b和f,g的适当假设下,运用喷泉定理和对偶的喷泉定理得到以上系统(*)的无穷多正能量解和负能量解.     

一类p-Laplace方程组的正径向解的存在性

本文研究p-Laplace方程组{-(rN-1(φ)(u'))'=λrN-1f(u,v), a＜r＜b,-(rN-1(φ)(v'))'=λrN-1g(u,v), a＜r＜b,u(a)=0=u(b),v(a)=0=v(b)的正解,其中参数λ＞o,(φ)是递增且同伦与R的奇映射,f,g∈[C[0,∞)]2满足适当的条件,讨论了当参数λ很大时正解的存在性.     

Infinitely Many Sign-changing Solutions for a Schr(o)dinger Equation in RN

We consider the existence of infinitely many sign-changing solutions for the nonlinear timeindependent schr(o)dinger equations of the form {-△u+Vλ(x)u=f(u), u∈H1(RN), (1.1) where Vλ(x) = λα(x) + 1.